Single and Multiple Emitter Localization in Cognitive Radio Networks

Transcription

1 Single and Multiple Emitter Localization in Cognitive Radio Networks by Suzan Ureten Thesis submitted to the Faculty of Graduate and Postdoctoral Studies In partial fulfillment of the requirements For the Ph.D. degree in Electrical and Computer Engineering School of Electrical Engineering and Computer Science Faculty of Engineering University of Ottawa c Suzan Ureten, Ottawa, Canada, 2016

2 Abstract Cognitive radio (CR) is often described as a context-intelligent radio, capable of changing the transmit parameters dynamically based on the interaction with the environment it operates. The work in this thesis explores the problem of using received signal strength (RSS) measurements taken by a network of CR nodes to generate an interference map of a given geographical area and estimate the locations of multiple primary transmitters that operate simultaneously in the area. A probabilistic model of the problem is developed, and algorithms to address location estimation challenges are proposed. Three approaches are proposed to solve the localization problem. The first approach is based on estimating the locations from the generated interference map when no information about the propagation model or any of its parameters is present. The second approach is based on approximating the maximum likelihood (ML) estimate of the transmitter locations with the grid search method when the model is known and its parameters are available. The third approach also requires the knowledge of model parameters but it is actually based on generating samples from the joint posterior of the unknown location parameter with Markov chain Monte Carlo (MCMC) methods, as an alternative for the highly computationally complex grid search approach. For RF cartography generation problem, we study global and local interpolation techniques, specifically the Delaunay triangulation based techniques as the use of existing triangulation provides a computationally attractive solution. We present a comparative performance evaluation of these interpolation techniques in terms of RF field strength estimation and emitter localization. Even though the estimates obtained from the generated interference maps are less accurate compared to the ML estimator, the rough estimates are utilized to initialize a more accurate algorithm such as the MCMC technique to reduce the complexity of the algorithm. The complexity issues of ML estimators based on full grid search are also addressed by various types of iterative grid search methods. One challenge to apply the ML estimation algorithm to multiple emitter localization problem is that, it requires a pdf approximation to summands of log-normal random variables for likelihood calculations at each grid location. This inspires our investigations on sum of log-normal approximations studied in literature for selecting the appropriate approximation to our model assumptions. As a final extension of this work, we propose our own approximation based on distribution fitting to a set of simulated data and compare our approach with Fenton-Wilkinson s well-known approximation which is a simple and computational efficient approach that fits a log-normal distribution to sum of log-normals by matching the first and second central moments of random variables. We demonstrate that the location estimation accuracy of the grid search technique obtained with our proposed approximation is higher than the one obtained with Fenton-Wilkinson s in many different case scenarios. ii

3 Acknowledgements I would like to express deepest gratitude to my supervisor Prof. Abbas Yongacoglu for his full support, expert guidance, understanding and encouragement throughout my study and research. Without his incredible patience and counsel, my thesis work would have been a frustrating and overwhelming pursuit. In addition, I express my appreciation to my co-supervisor Prof. Emil Petriu and also to Prof. Halim Yanikomeroglu for having served on my committee. Their thoughtful questions and comments were valued greatly. Thanks also go to the University of Ottawa which granted me an admission scholarship for the entire four year term. Without this scholarship, I wouldn t have been able to continue my studies on a full-time basis. Special thanks go to my colleagues, Dr. Zhenxia Zhang, Chao Li, Kerem Karatas and my other friends who helped me throughout this academic exploration. Finally I would like to thank my family for their patience, guidance, support and love in every aspect of my life. iii

4 Table of Contents List of Figures Nomenclature vi ix 1 Introduction Problem Definition Motivation Contributions Thesis Organization Publications Literature Survey Interference Map Generation RSS-based Single Emitter Localization Multiple Emitter Localization Interference Map Generation Radio Environmental Awareness (REA) andradio Environment Map (REM) RF Cartography Generation Techniques Global Interpolation Techniques Local Interpolation Techniques Based on Delaunay Triangulation Performance Evaluations of RF CartographyGeneration Techniques Network and Signal Model with Assumptions Performance Evaluations for RF Field Strength Estimation Emitter Localization Using the Generated Radio Environment Map Single Emitter Case Multiple Emitter Case iv

5 4 Improving Emitter Localization Accuracy Single Emitter Localization Cramer Rao Lower Bound for Emitter Localization Error Single Emitter Localization Using MLE (Grid Search) Method Computational Complexity of Grid Search Method Single Emitter Localization Using MCMC Method Multiple Emitter Localization Challenges of Multiple Emitter Localization Investigation into Sum of Log-normals A Sum of Log-normal Approximation from Simulated Data Multiple Emitter Localization Using Grid Search Comparison of the Proposed Approach with Fenton s Approximation 78 5 Complexity Reduction Iterative Grid Search Regular Iterative Grid Search Search Space Initialization Smart Initialization of MCMC Methods Conclusions and Future Work Conclusions Future Work Applying MCMC Techniques to Multiple EmitterLocalization Improving the Performance of MCMC Algorithms Estimating the Number of Emitters (Reversible Jump MCMC) RSS-based Localization with Different Case Scenarios APPENDICES 101 A Emitter Localization Methods for Different Case Scenarios 102 B Derivation of Likelihood Expression for Likelihood Based Estimators 103 B.1 Single Emitter Localization Problem B.2 Multiple Emitter Localization Problem C Quantization Error Calculation in Grid Search Method 108 References 112 v

6 List of Figures 3.1 Spatial interpolation An example of empirical and theoretical semivariograms Delaunay triangulation of a set of 10 nodes with the circumcircles Delaunay condition of triangulations (conditions satisfied) Delaunay condition of triangulations (conditions not satisfied) Delaunay triangulations of a random set consisting 20 points Delaunay triangulations of a random set consisting 100 points Delaunay triangulation and Voronoi tessellation of a set of 10 points Natural neighbor interpolation using Voronoi tessellations Correlated shadow fading map with spline interpolation RF field strength estimation at output sites using input sites Voronoi tesselation and convex hull of input sites Efficiency of RSS estimates in correlated shadowing vs d corr Efficiency of RSS estimates in correlated shadowing vs N s Efficiency of RSS estimates in correlated shadowing vs d corr (Delaunay) Efficiency of RSS estimates in correlated shadowing vs N s (Delaunay) Localization errors of cartography generation techniques vs db spread Localization errors of cartography generation techniques vs N s Localization errors of cartography generation techniques vs db spread Localization errors of cartography generation techniques vs N s Nearest neighbor interpolation with multiple emitters Natural neighbor interpolation with multiple emitters Linear interpolation with multiple emitters Natural neighbor interpolation with high location estimation error Delaunay triangulation formed with 100 sensors vi

7 3.26 Prob. of RSS estimation for 5 db for nearest n. interpolation Prob. of RSS estimation for 10 db for nearest n. interpolation Prob. of RSS estimation error for 5 db for natural n. interpolation Prob. of RSS estimation error for 10 db for natural n. interpolation Prob. of RSS estimation error for 5 db for linear interpolation Prob. of RSS estimation error for 10 db for linear interpolation Comparison of RSS estimation error probability for 5 db vs. db spread Comparison of RSS estimation error probability for 10 db vs. db spread Prob. of RSS estimation error for 5 db for linear interpolation vs N s Prob. of RSS estimation error for 10 db for linear interpolation vs N s CDF of RSS estimation error for linear interpolation Comparison of RSS estimation error probability for 5 db vs N s Comparison of RSS estimation error probability for 10 db vs N s Distance and angle for parameter G in calculating CRLB Uniform sensor geometry Random emitter sensor geometry CRLB for uniform and random emitter-sensor geometry CRLB for uniform sensor placement for all emitter locations Uniform grid structure for full search Sensors positioned on the circle centered at the emitter Performance of full grid search compared to CRLB for 500 m radius CRLB and RMSE of the grid search for uniform sensor placement Computational complexity of the MLE method for Q = An example for Accept-Reject technique, uniform envelope An example for Accept-Reject technique, scaled Gaussian envelope A simulation example: Single emitter localization using MCMC RMSE performances of MH and interpolation based techniques RMSE performance vs number of MH iterations for different step-sizes RMSE performance vs number of burn-in iterations for different step-sizes Examples of log-normal density functions Likelihood difference between log-normal and gamma approaches Likelihood difference between log-normal and Gaussian mixture approaches 77 vii

8 4.20 Localization error for scenario Localization error for scenario Localization error for scenario Localization error for scenario An example of 3 level 2x2 partitioning for iterative grid search Full grid search performances with different grid sizes Performance comparison of full and iterative grid search vs db spread Performance comparison of full and iterative grid search vs N s Performance comparison of regular, maximum and minimax approaches Performance of maximum approach vs db spread Performance of minimax approach vs db spread RMSE of the smartly initialized MCMC techniques vs db spread RMSE of the smartly initialized MCMC techniques vs N s C.1 Quantization error calculation in grid search for region C.2 Quantization error calculation in grid search for region viii

9 Nomenclature Abbreviations AOA Angle of arrival CPSO Constriction particle swarm optimization CR Cognitive radio CRLB Cramer Rao lower bound CRN Cognitive radio network CWLS Constrained weighted least squares EM Expectation maximization FIM Fisher information matrix GDOP Geometric dilution of precision GIS Geographic information systems GPS Global positioning system GSM Global system for mobile communications IEEE Institute of electrical and electronics engineers IM Interference management LS Least squares MAC Media access control MCMC Markov chain Monte Carlo ME Mean error MH Metropolis-Hastings MH-Linear Linear interpolation initialized MH MH-Max Maximum power reading sensor s location initialized Metropolis-Hastings MH-NN Nearest neighbor interpolation initialized Metropolis-Hastings MH-RND Randomly initialized Metropolis-Hastings ML Maximum likelihood MLE Maximum likelihood estimation MMSE Minimum mean squared error MSE Mean squared error PHY Physical layer REA Radio environmental awareness REM Radio environment map RF Radio frequency RIFE Radio interference field estimation ix

10 RJMCMC Reversible jump Markov chain Monte Carlo RMS Root mean square RMSE Root mean squared error ROI Region of interest RRM Radio resource management RSS Received signal strength RSSD Received signal strength difference TDOA Time difference of arrival TOA Time of arrival TSA Tree search algorithm WSN Wireless sensor network WTLS Weighted total least squares x

11 Mathematical Symbols α β ˆθ θ c θ i θ new θ p θ ǫ η γ(.) κ L L c L p L R L Fenton L fitted L mar λ φ(.) σ M r d j d corr E(.) E rms G H I J ij L LN(.) ln( L) N s N t N selected P t Q r j path-loss exponent constant that reflects carrier frequency and antenna gain (ˆT x, ˆT y ) emitter s estimated location current state of location parameter initial state of location parameter new state of location parameter proposed state of location parameter (T x,t y ) emitter s actual location ln10 10 efficiency semivariance function RMSE of emitter localization likelihood function likelihood value of current location likelihood value of proposed location likelihood ratio likelihood value of Fenton approximation likelihood value of fitted approximation marginalized likelihood function wavelength basis function shadowing spread signal model vector of RSS measurements distance between jth sensor and the emitter decorrelation distance statistical expectation operator root mean squared value of received power estimation error geometric dilution of precision number of simulations number of iterations (levels) Fisher information matrix length of square search area log-normal operator log-likelihood function number of sensors number of emitters number of selected sensors emitter power number of grid elements received power at jth sensor xi

12 s j total mean received power at jth sensor SLN(.) sum of log-normal step size of MH algorithm µ j mean value of sum-of-log-normal distribution σsum 2 j variance of sum-of-log-normal distribution a acceptance ratio d ij distance between jth sensor and ith emitter r ij power at jth sensor due to ith emitter mean power at jth sensor due to ith emitter s ij xii

13 Chapter 1 Introduction An increasing number of wireless systems and applications are being integrated into our daily lives in a variety of forms, such as entertainment devices, defense, security applications and health care systems. Any such wireless system needs radio frequency (RF) spectrum allocation for its proper operation, therefore spectrum scarcity has become a critical issue due to the increased demand for wireless systems and applications. The RF spectrum is considered as a natural source and traditionally its use has been strictly regulated and managed by government on a licensing basis. These strict regulations do not allow the use of spectrum by other radio technologies or different users other than the specific ones initially defined, i.e. license holders are not allowed to reallocate the spectrum to different technologies or to other users. However, several spectrum measurement campaigns carried out in several places have demonstrated that major licensed bands such as those allocated for television broadcasting, amateur radio or paging systems are under-utilized, i.e. the allocated spectrum is idle at different times and geographical locations. Therefore it is believed that the existing problem is the inefficient utilization of the spectrum due to the current regulation scheme rather than the scarcity of the spectrum. The concept of cognitive radio has been recently proposed to address the under-utilization problem due to current spectrum regulation regime. A cognitive radio is a device that can change the transmit parameters dynamically based on the interaction with the environment it operates. The objective of the cognitive radio is to achieve better spectrum utilization by allowing unlicensed users to access the spectrum if their usages do not cause unacceptable interference to the original license holders. In practice, the unlicensed users, also called secondary users or cognitive users, need to sense the spectrum continuously to find a suitable spectrum band for possible utilization without causing interference to the licensed users, also called primary users. Spectrum sensing task is a critical part of a dynamic spectrum access system and complicated by several factors such as requirement of detecting the primary user at very low signal-to-noise ratio levels, measurement uncertainties, shadow fading and hidden terminal problems. Errors in sensing can cause performance degradation both in the primary or the secondary user systems. When the secondary users misses detecting the primary user (called missed detection) and transmits, it causes interference to the primary user. On 1

14 the other hand, if the sensing result indicates presence of the primary user when the spectrum is actually idle (called false alarm), then a transmission opportunity is lost and secondary user efficiency suffers. A promising idea to address these difficulties is to perform sensing collaboratively among several radios in the network. In order to assist cognitive radio to perform these tasks, a concept called radio environment map (REM) is proposed. The REM is a tool to provide network support to cognitive radio. It identifies the location of transmitters and geo-spatial properties of parameters such as terrain, service availability, policy requirements and hardware type. It contains several kinds of information such as geographical structure of the area and the spectrum regulations in the area or the location of transmitters. If the locations and the transmit powers of primary users can be extracted from such tools, then a protection region can be set around the primary emitters. This protected region is known as primary exclusion region or zone and primary users are protected from harmful interference due to secondary users inside this region. This type of approach will help efficient spectrum exploitation which is an essential step in dynamic spectrum access networks. Therefore there exists a need to identify the location of primary emitters to predict the primary exclusion region as the emitters cannot or will not identify their own location and/or communicate it to a REM due to non-cooperative nature of primary networks. Spatial estimation of the location of signal sources, or source (node) localization problem has been considered extensively in the literature where the goal is to locate the position of a source based on the measurements taken at the sensors. Source localization has several applications in various fields such as logistics and tracking [54]. For example, accurate localization of cell phones in an emergency has been mandated in the United States [89]. Similarly, microphone arrays can be used to determine the location of an acoustic source [11] or to aid automatic camera tracking [77]. In source localization, received power measurements can be utilized to estimate the distances between the source and the sensors by using a protocol in which the source with unknown location transmits a signal with known power and the sensors deduce their distance from it based on the received signal power measurements and a given path-loss model. The distances from several monitors are then combined to obtain an estimate of the relative location of the transmitter [18]. The problem can also be extended to a network of multiple sensors whose locations must be determined based on the known locations of a group of nodes and pairwise node distances [6]. The literature on source localization has been developing on efficient and/or distributed algorithms for minimizing the localization error. An important constraint of source localization problem is that the source (to be localized) is not cooperating and its transmit power is unknown; this makes it infeasible to compute the distance between the source and the monitor. The problem is even more complicated when there are multiple sources to localize and their number is unknown. Measurements for source localization include received signal strength (RSS) [11], angle of arrival (AOA) [66] and time difference of arrival (TDOA) [14] at multiple receivers. The drawbacks of AOA measurements is that determining the AOA requires a phased array antenna at each sensor rather than a single antenna and the quality of the final position estimate degrades rapidly as the monitors move away from the source node. Due to its simplicity in terms of both hardware and computational requirements, RSS is frequently 2

15 used and is an attractive source of measurements with wide applicability for low-cost sensor systems. However, RSS measurements are not very accurate and a large, dense network is often required for precise location estimates. RSS techniques typically assume that the transmitted power and the path-loss exponent are known and the transmitting antenna is isotropic. TDOA methods do not make such assumptions, but they require significantly higher communications overhead than AOA or RSS-based methods. In addition to the existing localization methods, fingerprinting based localization is also an attractive source of RSS based measurement. It simply relies on the recording of the signal strength from several access points in range and storing this information in a database along with the known coordinates of the client device in an offline phase. This information can be deterministic [8] or probabilistic [81]. During the online tracking phase, the current RSS measurement at an unknown location is compared to those stored in the fingerprint and the closest match is returned as the estimated user location. Its main disadvantage is that any changes of the environment may change the fingerprint that corresponds to each location, requiring an update to the fingerprint database. However, the integration with other sensors can be used in order to deal with the environmental interference [10]. Emitter localization problem in general has been considered extensively in the literature; see for example [43] for an overview of localization techniques. Localization may be accomplished via the above mentioned techniques. Among these techniques, less accurate RSS-based techniques are often of interest as they require simpler hardware. The challenge of RSS-based localization is due to numerous factors affecting the energy decay between the transmitter and emitter such as shadowing, multipath, path-loss exponent estimation errors, geometric configuration of the nodes and antenna orientation. Despite having several sources of error, RSS-based techniques are expected to perform satisfactorily when a large number of spatially separated sensors are employed. 1.1 Problem Definition In this thesis, we consider the problem of generating an interference map of a given geographical area, and estimating the locations of multiple radio transmitters that operate simultaneously in adjacent coverage areas by using the received signal strength measurements taken at different locations. We develop a probabilistic model of the problem and propose algorithms to address location estimation challenges. The efficiencies of the algorithms are also of interest, hence we explore complexity-performance trade-offs as well as the distribution of computations over the network. A mathematical definition of the problem including model assumptions and constraints will be presented in Section and Section for single and multiple emitter case scenarios respectively. The assumption of model parameters will be stated for different localization techniques in each related section. However, a table that summarizes the assumptions of model parameters can also be found in Appendix A. 3

16 In this chapter, we will justify the motivation of our efforts and briefly review the existing literature. 1.2 Motivation The problem of locating and tracking signal-emitting sources has attracted attention for the last 60 years [21]. Early applications in radar and sonar typically involved a few sensors. In the last six decades, there has been a considerable increase in the sophistication of wireless networks. Dramatic advances in radio frequency have been made possible through the use of large networks of wireless sensors for a variety of new monitoring and control applications. These advancements have led to the broadening of techniques employed for localization as well as the applications where localization is important. Examples of today s applications for source localization that serve as the major reason for current research include the following: Monitoring and tracking for safety and security reasons [62] Location based billing [62] Fraud protection [62] Asset tracking [62] Fleet management [62] Alternative to global positioning system (GPS) navigation [62, 21] For outdoor localization, GPS systems have been used in a wide range of applications including tracking, transport navigation, and guidance [32]. Although GPS works extremely well in outdoor localization, it does not perform well indoors. Therefore alternatives to GPS navigation is being sought to overcome its limitations. When inside large buildings or in an underground or a tunnel, there is often no direct line from the satellite signals to the GPS receiver and the signal weakens or distorts as it travels through the building. The GPS works better when the device has a clear line of sight to the sky. The more GPS satellites that a device can access, the more accurate location estimates it provides. The construction materials in a building also affect how well a GPS device will work indoors. GPS signals pass through glass much more easily than they do through thick, solid materials such as brick, metal, stone or wood. For example, if one is in a house and stand near enough to the windows, or in an office tower with large windows, the device could still work. Spatial estimation of the location of signal sources, or source (node) localization is already discussed in the Introduction Section with some other applications. Besides source localization, spectrum cartography, i.e. multiple transmitter localization and communication footprint identification has also applications in cognitive radio white space networking 4

17 and spectrum enforcement. An accurate and fast method of obtaining up-to-date transmitter footprint information is necessary for frequency based networking and optimal routing of cognitive radios, while ensuring minimal interference to higher priority primary users. Moreover, recent studies have revealed that knowledge of the primary users location and interference tolerance can dramatically improve spectrum sensing performance, and therefore reduce interference to the primary system [35]. For example, once the location of the primary emitter is known, the secondary user that is the closest to the emitter can detect either the primary is on or off. We consider the problem of using RSS measurements taken by the secondary users to estimate the locations of multiple primary transmitters that operate simultaneously in adjacent coverage areas. We develop a probabilistic model of the problem and propose algorithms to address location estimation challenges. The proposed algorithms provide a mechanism for secondary users to obtain their information about nearby primary systems by processing distributed measurements. The efficiency of the algorithms is also of interest, and hence we explore complexity-performance trade-off as well as the distribution of computations over the network. Localization of multiple emitters can also be applied to dynamic spectrum access in cellular networks [50] where the possibility of deploying a cognitive network that can opportunistically access the spectrum allocated to a primary cellular network is studied. The main problem is that detecting spectrum opportunities in the frequency bands allocated to cellular networks is more challenging than in TV white space due to several problems in the former, namely the requirement for coverage, the presence of different services with different quality constraints, dynamic traffic patterns, the presence of several neighboring primary transmitters and adaptive primary transmit powers. The approach in [50] assumes that the cognitive network knows the positions of the primary base stations and its own base stations through a local or regional database without knowing their transmit powers. It utilizes a maximum likelihood estimator (MLE) to detect the activity of neighboring primary base stations and exploits this information to determine the power with which a cognitive radio can transmit. The results show the possibility of using such a method to deploy an IEEE like network as a cognitive network under specific constraints related to the position, to the number and to the distribution of the spectrum monitoring sensors inside the primary network. Another challenge for opportunistically accessing the spectrum in cellular networks is that the secondary users need to detect a wide-band of 20 MHz spectrum, a typical bandwidth of a GSM network so that they can decide whether the primary user is on or off. This requires wide-band power spectrum estimation which is highly costly. Besides, distinguishing channels allocated to neighboring cells and to the own cell of a secondary user requires FFT analysis and fast signal processing algorithms that would cause latency in the cognitive network. Instead, low-cost sensors that monitor the activities of primary base stations can be distributed in the network and the measurements collected at these sensors can then be sent to a fusion center where the information is utilized to localize multiple primary base stations. When there are multiple base stations transmitting in the same band, energy detection techniques are insufficient to localize the base stations that are transmitting simultaneously. This is because each base station contributes different unknown portions of power at the sensor nodes. Simple methods based on received power 5

18 measurements are inconclusive. Isolating each base station based on the its signature is essential to estimate its location in this scenario. There is a trade-off between the cost and complexity of the sensors. When we require complex signal processing algorithms in the sensors, the cost of the devices increases. Thus, there is a need for signal processing and detection algorithms which can be implemented at low cost. The time taken for localization directly impacts the time to find the white spaces in the spectrum. In highly dynamic networks like cellular networks where white spaces do not last long, time to localize needs to be short so that secondary users can make better use of white spaces. Hence algorithms that take less time to localize are preferable in cognitive radio networks. The possibility that REMs can also be used as databases that provide a variety of network and user-related context information for improving interference management (IM) and radio resource management (RRM) in two-tier cellular networks comprising macro and femto-cells [83] is another motivation for us to study emitter localization problem in cognitive radio networks. A cognitive femto-cell is a femto-cell that intelligently and dynamically allocates air interface resources based on the usage of nearby macro-cells [33]. The framework is essentially a dynamic spectrum access framework in which the macro-cell is the primary user and the femto-cell is the secondary user. The femto-cell senses what time-frequency resources are in use by the macro-cell and dynamically allocates resources such that it does not interfere with the macro-cell s transmissions. Dynamic allocation offers greater flexibility and potentially improved spectral efficiency over fixed allocations. Given the strong interest in femto-cells in the mobile networking industry, it is likely that cognitive femto-cells will represent a key use of dynamic spectrum access technologies in coming years. Another potential application of dynamic spectrum access in a mobile cellular network is device-to-device communication. Device-to-device communication is a direct mode of communication between nearby mobile devices that would otherwise be routed through the base station and associated infrastructure. It is proposed to offload traffic from potentially congested infrastructure as well as to provide communication opportunities in the absence of a link to the base station (e.g. indoor scenario). Commercial networks, such as media sharing and gaming that offers proximity based services, have interest in device-to-device communication. There is also interest in non-commercial networks. For example, the public safety network, such as FirstNet (First Responder Network Authority) that is designed to serve the needs of the first responder community, uses direct mode communication between emergency responders on the network [61]. As the first responder community migrates to broadband cellular networks, a device-to-device capability is an active research topic in order to preserve this critical mode of communication. The 3rd Generation Partnership Project (3GPP) has studied the feasibility of proximity-based services in preparation for inclusion in Release 12 of Long Term Evolution (LTE) specifications [38]. Since device-to-device connections can be controlled by the cellular network infrastructure [22], a framework in which the connections are established in a decentralized fashion by the devices themselves is also possible. Such a framework would allow for a device-to-device communication in the absence of a link to the base station, which is highly desirable in the public safety community. In the decentralized scenario, the peerto-peer network would function as a secondary network, sensing for and giving priority to 6

19 the primary transmissions. Dynamic spectrum access can also play an important role in unlicensed spectrum. As more devices appear on unlicensed bands, it may become necessary to prioritize traffic. For example, it is common to have multiple WiFi devices in a home connected to the internet through a single access point. Currently, they access the medium as peers. But it may be more appropriate to set priorities among the devices. For example, the device running a video streaming application may have a higher priority than the smart fridge reporting its temperature. Using dynamic spectrum access technology, the smart fridge can transmit its packets when it detects white space in the spectrum, rather than competing with the video streaming application. Furthermore, by sharing information about the radio environment through REM dissemination, the hidden node problem can be mitigated and the secondary users can co-exist with primary users with minimal harmful interference [85]. 1.3 Contributions This thesis makes 4 major contributions: A number of interpolation techniques have been used for generating interference cartographs in the literature but there is no such relative performance comparison of these techniques where the data to be interpolated are the RF field strength values. As one of our contribution to the area of interference map generation problem in CR networks, we present a comparative performance evaluation of the interpolation techniques used in interference cartography generation. Specifically, we investigate the performance of Delaunay triangulation based interpolation techniques in generating interference maps as the use of existing triangulation provides a computationally attractive solution for the map generation problem. Furthermore, we propose to estimate the locations of the primary emitters from the generated interference map and explore complexity-performance trade-offs of these interpolation techniques. As the accuracy of the estimation obtained from interference maps is low, we investigate more accurate algorithms, develop a probabilistic model of the problem and propose algorithms to address location estimation challenges. The existing maximumlikelihood estimation (MLE) based on grid search method gives accurate location estimations with a constriction of unknown emitter power for the single emitter case scenario. However, since the grid elements must be chosen small to obtain more accurate location estimations, this estimator requires a high number of computations. Therefore we propose an efficient RSS- based localization algorithm, referred to as iterative grid-search method that closely approaches the MLE method, particularly at small shadowing spread values with significantly reduced computational complexity. As an alternative to MLE-based grid search algorithms, we propose to apply a Markov chain Monte Carlo (MCMC) based technique to the emitter localization 7

20 problem. The main reason for this is, MCMC based techniques provide lower computational complexity solutions to parameter estimation problems especially when the number of parameters to be estimated is large. To the best of our knowledge, MCMC techniques have not been applied to multiple emitter localization in the literature. The major drawback of these techniques is that they require a large number of iterations to converge if the design parameters of the algorithm, such as burn-in samples and step size are not tuned in advance. To overcome this problem, we propose to initialize the algorithm with the knowledge fed-back from the interference map generated by using low-complexity interpolation techniques. Hence the smart initialization of MCMC technique we propose, eliminates tedious parameter tuning process and achieves significantly better localization performance than randomly initialized ones at a fraction of iterations. The grid search technique used for multiple emitter localization requires the knowledge of the pdf of the sum of log-normal distributions which does not have an exact closed form. In this work, we use the Fenton-Wilkinson approximation due to its simplicity and applicability to non-iid log-normal random variables that represent different mean received power values for each emitter sensor pair in the network. However, the estimation accuracy of this approach is low at high db spread values. Hence we propose a new approximation where the pdf of sum of log-normals is estimated from a set of simulated data. Our proposed approach addresses the problem with the Fenton-Wilkinson approach and gives higher estimation accuracies, even at high db spread values. This approach is our final contribution to the area of RSS based multiple emitter localization. 1.4 Thesis Organization The outline of the thesis is as follows: Chapter 2 provides a literature overview for interference map generation, RSS-based localization, such as maximum likelihood estimation and iterative grid search techniques, Cramer Rao lower bound (CRLB) for RSS-based localization and also in multiple emitter localization. Chapter 3 explains the importance of environmental awareness and the use of REM in cognitive radio networks followed by a brief overview of the interpolation techniques considered in generating the interference map. The RF cartography generation techniques studied in this thesis include global interpolation techniques such as thin-plate spline and Kriging interpolation and local interpolation techniques that are based on Delaunay triangulation. Followed by the definition of the probabilistic signal model and network assumptions, a comparative performance evaluation of RF cartography generation techniques in terms RF field strength estimation and emitter localization accuracy for both single and multiple emitter case scenarios is given in this chapter. Chapter 4 investigates the emitter localization problem in detail with the aim of improving the location estimation accuracy obtained from the generated inference map and 8

21 presents a framework that leads to a proposed approach to address the multiple emitter localization problem. Performance analysis of the existing MLE based on full grid search method isgiven andits performance iscompared with Cramer Rao lower boundfor a single emitter case. Remarking the problems with the grid search method, the chapter continues with a study of proposed MCMC based localization technique with its performance analysis. After summarizing the challenges of multiple emitter localization and the investigation into sum of log-normal random variables, the chapter concludes the study of multiple emitter localization problem by proposing a novel approach called pdf estimation approach from simulated sum of log-normals. The simulation results that demonstrate the performance evaluations of grid search method with the proposed approach and Fenton s approximation are also presented. Chapter 5 explores the complexity reduction of the localization algorithms studied in the thesis. It presents different iterative grid search techniques and smart initialization of MCMC techniques with their performance analysis. Chapter 6 concludes the thesis and points out open problems for future work in the related area. 1.5 Publications The related publications are as follows: S. Ureten, A. Yongacoglu and E. Petriu, A comparison of interference cartography generation techniques in cognitive radio networks in Communications (ICC), 2012 IEEE International Conference on, pages S. Ureten, A. Yongacoglu and E. Petriu, Interference map generation based on Delaunay triangulation in cognitive radio networks in Signal Processing Advances in Wireless Communications (SPAWC), 2012 IEEE 13th International Workshop on, pages S. Ureten, A. Yongacoglu and E. Petriu, Primary emitter localization using smartly initialized Metropolis-Hastings algorithm in Signal Processing Conference (EUSIPCO), 2013 Proceedings of the 21st European, pages 1-5. S. Ureten, A. Yongacoglu and E. Petriu, A reduced complexity iterative grid search for RSS- based emitter localization in Communications (QBSC), th Queens Biennial Symposium on, pages S. Ureten, A. Yongacoglu and E. Petriu, Iterative grid search for RSS- based emitter localization in Signal Processing Conference (EUSIPCO), nd European Signal Processing Conference, pages

22 Chapter 2 Literature Survey In this chapter we review the literature on interference map generation, RSS-based geolocation algorithms and multiple emitter localization in cognitive radio networks. 2.1 Interference Map Generation Cognitive radios (CRs) are goal-oriented and aware of their operational environment, learn from experience and adopt themselves to changing operating conditions. The radio environment map (REM) concept has been proposed to support their operations. The map itself is an integrated database that characterizes the radio environment for CRs. It represents the radio environment in multiple domains such as geographical features, available services, spectral regulations, locations and activities of radios, relevant policies and experiences. In [87], a cost-efficient approach is proposed to improve spectrum utilization in both space and time domains by using the REM-enabled CRs as secondary users. The paper also evaluates the performance of spectrum sharing wireless networks when CR uses the global REM and the local REM. In [85], it is shown that the REM can be exploited by the CRs for various cognitive functionalities such as situation awareness, reasoning, learning, planning and decision support. The authors present the system flow and framework of REM-enabled situation aware learning algorithms in the paper. In [7], a prototype of a REM for storing and reasoning about spectrum data obtained from heterogeneous sources is described and shown that information such as transmitter locations and estimates of spectrum occupancy over space and time can be obtained through the REM. In [2], interference cartography concept is presented in secondary spectrum usage. An interference cartograph is a map that displays the level of interference over the area of interest and it can constitute one of the layers of REM. Interference cartography involves combining measurements coming from different radio network entities together with the geo-location information and applying effective spatial interpolation techniques to obtain a map which indicates interference levels at each grid point over the area of interest. Utilization of interference cartography in a secondary spectrum usage permits the secondary network to be aware of local interference tolerance levels so that it becomes possible to 10

23 detect, identify and use spectrum opportunities without disturbing the primary users or other secondary users in the network. A number of spatial interpolation techniques have been used for generating interference cartographs in the literature. In[2], Kriging is applied on spatial interference data obtained from a radio network simulator as a simple and efficient spatial interpolation method. The original interference map is obtained from partial knowledge of interference data points. Different percentage values of partial knowledge are investigated and the results demonstrate that interference cartography constitute a viable solution for efficient secondary spectrum usage. In [59], the Kriging interpolation technique is adopted to estimate the spatial power spectral density. An iterative REM building process based on Kriging interpolation is presented in [30]. The thin-plate spline interpolation technique is employed to accommodate shadow fading in spectrum cartography generation [46]. There have been many different interpolation techniques studied in the interference cartography literature. For example, the techniques used in the above reviewed papers are the global interpolation methods. The local interpolation techniques which require data only from the neighboring nodes have also been studied recently. Specifically, triangulation is a generic tool that serves as a basis for many geometry-based algorithms in wireless networks to enable local information processing. Delaunay triangulation is the one that has been frequently used for many networking tasks such as routing [12] and topology control [40]. The interpolation technique based on Delaunay triangulation is used to generate an interpolated map grid as a simple and a fast solution in the context of networks in [23]. One of the most challenging tasks in interference cartography through interpolation is choosing the most appropriate technique as the validity of each method depends on the type of data being interpolated. The importance of the evaluation of the relative performances of interpolation techniques has been recognized in various fields, including remote sensing [25], image reconstruction [41] and wireless sensor networks [69]. However, there is no such relative performance comparison of the interference cartography generation techniques, where the data to be interpolated are the RF field strength values. The RF field strength estimation has also been studied in radio propagation prediction literature, where the objective is to predict the field strength values in the region of interest (ROI) for a given emitter location and power information. In the context of CR networks, however, the desire is to estimate the location and power of the primary emitter in addition to estimating the field strength values at any location in the ROI using a limited number of field strength measurements. In this thesis, we provide a comparison of the performances of some common interpolation techniques studied in literature. Our goal is to provide a comparative analysis of the techniques in terms of RF field strength estimation and emitter localization for single and multiple emitter case scenarios. 2.2 RSS-based Single Emitter Localization The literature on RSS-based primary emitter localization has been on developing efficient algorithms for accurate location estimation. An important constraint of primary emitter 11

24 localization problem is that the primary user is not cooperating and its transmission power is unknown. This limitation is critical in localizing primary users because most schemes using RSS measurements require transmission power of the RF signal. In [36], the primary emitter power and location estimation are modeled as a constrained optimization problem with a weighting factor. The proposed scheme uses the linearization technique to approximate the relationship between RSS measurements and unknown power and coordinates of the primary user to set the weighting factor that considers the differences of the quality of measurements. This approach is shown to outperform the least squares technique. In [44], a scheme is proposed to characterize the spatial size of the perceived spectrum hole in terms of the maximum permissible transmit power using signal measurements taken by a group of frequency agile radios. In [76], single source localization problem with unknown transmit power is studied by using an approach that approximates the ML problem to a convex optimization problem. The proposed approach is shown to have remarkable performance close to ML estimator. Linearizing RSS model, least squares (LS) and weighted total least squares algorithms (WTLS) are also derived for the problem in [76] and it is shown that WTLS has a better performance than LS. Another study for RSS linear estimators is reviewed in [16]. In [16], a constrained weighted least squares (CWLS) positioning approach that encompasses different measurement cases such as TDOA, AOA and RSS is presented. It is shown that CWLS location estimators can achieve the Cramer-Rao lower bound when measurement error variances are small. In addition to the mentioned localization techniques, maximum likelihood technique is one of the most frequently used method in the literature. Among the others, ML estimation offers an attractive approach for localization problems since it is asymptotically efficient, unbiased and does not require any prior information of the transmit power [65]. In fact, it has been shown that MLE method achieves the Cramer-Rao lower bound (CRLB) at small shadow variances [80]. However, because of non-convex behavior of ML estimator, intensive computations are required to achieve its global minimum. In [55], the ML estimator and the CRLB are derived for self-localization of a network of sensors, in which a small subset of the sensors are anchor nodes at known locations. In [82], a maximum likelihood based approach is given and two numerical solutions are proposed based on simplex optimization and grid search. In my thesis, I study the grid search technique as a numerical solution to the MLE method for locating a single emitter with unknown emitter power and for locating two emitters with known and equal emitter power values. In a grid search approach, the algorithm scans all the possible grid points in the localization space. The grid point that maximizes the likelihood is selected as the location of the emitter. The size of the grid elements must be chosen small to obtain more accurate location estimates. However, smaller grid size increases the computational complexity. In order to reduce the complexity, different iterative grid search techniques are studied in the literature. For example, a variable-mesh, derivative-free optimization algorithm, namely contracting grid search method, is used to derive interaction locations in compact gamma cameras in [31]. In [34], the location of a sound source in a distributed sensor network is estimated using a grid based multi-resolution search to reduce the complexity of an exhaustive maximum likelihood estimator and a smarter multi-resolution search is proposed based on searching around the highest energy reading sensor. [56] proposes a low-complexity positioning 12

25 procedure that simply searches for the global minimum around the sensor exhibiting the smallest local maximum of the cost function and it is shown that it outperforms the naive approach that searches for the global minimum around the sensor reporting the largest signal strength. In [3], a tree search algorithm (TSA) is used to reduce the computational complexity of grid search algorithm in sensor networks assuming that the power of the transmitter to be located is known and it is shown that the performance of the TSA algorithm closely achieves the performance of least squares estimator with significantly reduced computational complexity. In my thesis, I use three different iterative grid search techniques and present their performance analysis in localizing a single emitter with the assumption of unknown emitter power. Our regular iterative algorithm sets a search space centred at the origin of the ROI and it does not rely on any sensor measurements or sensor measurement related metric in setting the initial search space. However the other two approaches rely on sensor measurements with the rational given in [34] and [56] for setting the initial search space. 2.3 Multiple Emitter Localization Multiple emitter localization and communication footprint identification are recent topics of research. In [48], the location and average power estimation problem of two transmitting sources which is based on the measurements of the power received by sensors that are placed in known locations in a given area is studied. A technique for obtaining radio interference field estimation(rife) by estimating average transmit power via maximizing the likelihood function (ML criterion) in a shadow fading environment is presented. Simulated annealing technique is used for maximization. The effect of number of sensors, grid size, shadowing variance and additive noise variance on the averaged estimated power are investigated. It is shown that increasing the number of sensors beyond certain level does not result in higher estimation accuracy. The estimated power on the actual source locations is also shown to be decreasing when the shadow fading variance increases. In [82], blind estimation (i.e. estimation without any prior knowledge about the transmitters location or any statistical characterization of its transmit power) of a wireless node s transmit power based on received power measurements at multiple sensor nodes is studied. First, through geometrical analysis, it is shown that there is a fundamental limitation on uniquely estimating the unknown transmit power and location of a node for certain topologies of sensor placements even though infinite measurements are considered. In other words, regular placement of monitors in the ROI with uniform spacing does not provide sufficient measurement diversity to yield a unique estimate. Second, analytical expressions for the maximum likelihood estimator under log-normal fading model are obtained. Third, it is shown that the ML estimator is asymptotically optimal for a random topology of monitor placement, i.e. the transmit power estimate converges to the actual value as the number of sensors goes to infinity. In [52], the number of transmitters and their transmit powers are assumed to be known, and a global optimization approach for locating multiple transmitters within a geographical area is proposed. A set of sensor nodes are assumed to be present in the region to 13

26 measure the total received power at their respective locations. These measurements are sent to a processing node, which uses particle swarm optimization to find the transmitter locations that minimize the difference between the true received power and the estimated power values at the sensor locations. Clustering is used to generate initial estimates of the transmitter locations, thereby increasing the likelihood that the particle-based optimizer reaches the global minimum. Simulation results show that global optimization is an effective method for multiple transmitter localization and that generating smart initial conditions via clustering improves the performance. In [51], estimating multiple transmitter locations based on received signal measurements of randomly located receivers under log-normal shadowing with quasi expectation maximization (EM) technique is studied. Simulated performance of quasi EM algorithm is compared to random guessing and global optimization using constriction particle swarm (CPSO). Results showed that the proposed quasi EM algorithm outperforms the alternative methods, especially as the number of transmitter increases. In this work, the number of transmitters is assumed to be known and all the transmitters are assumed to have the same constant transmit power. In [49], the number of transmitters is estimated by minimizing the sum of mean squared error (MSE) in the location and power estimates. The two main criteria are presented to determine the total number of transmitters in the primary system. The first criterion is called the net minimum mean squared error (MMSE) criterion, which uses the Cramer- Rao lower bound on localization accuracy. The second criterion is an information theoretic criterion namely minimum description length. Both of these criteria lead to measurement clustering algorithms in a natural way. Although only signal strength measurements are considered, the approach in [49] is predicted to be generalized to include other types of observations (e.g., time and angle information) with independent measurements in additive noise. The numerical results demonstrate the effectiveness of the proposed approach to measurement clustering. In [88], an iterative multiple primary user localization algorithm which estimates the positions non-cooperatively based on k-mean clustering is proposed and its performance is compared with the traditional EM algorithm. Here, similar to [51], the number of transmitters is assumed to be known and their power values are assumed to be equal. The received signals by sensing nodes from each primary user are assumed to be uncorrelated. Authors claim that the proposed iterative method achieves better performance than the EM algorithm where the proposed iterative method has lower computational complexity than EM algorithm as the EM requires some complex matrix operations. All of these methods are based on transmitting the RSS measurements from multiple sensors to a central node prior to location estimates. Compared to other physical layer measurements, such as the time of arrival and angle of arrival of radio signals, measuring the received power is simple in terms of both hardware and computational requirements, and thus, it is an attractive source of measurements with wide applicability for low-cost sensor systems. In the thesis, we extend our studies of grid search technique to the localization of multiple emitters by using the RSS measurements collected at multiple sensors. The MLE algorithm requires the knowledge of the pdf of the sum of log-normal random variables 14

27 that represent the portion of received power values for each emitter sensor pair in the network. However, the sum does not have an exact closed form. Therefore approximations for sum of log-normal random variables are of interest for emitter localization methods, such as MLE, EM andmcmc methods that areutilizing therss measurements at sensors for estimation. We have also investigated the existing sum of log-normal approximations in Section and proposed a novel approach that improves the performance of our multiple emitter localization algorithm compared to the existing Fenton-Wilkinson s [64] sum of log-normal approximation approach. As an alternative to MLE-based grid search algorithms, we also study the Metropolis Hastings (MH) algorithm which is an MCMC based technique to the emitter localization problem since MCMC based techniques provide a lower computational complexity solutions to parameter estimation problems especially when the number of parameters to be estimated is large. To our best knowledge, MCMC techniques have not been applied to multiple emitter localization in the literature. Smartly initialized MH algorithms are our additional contributions to the area of RSS based emitter localization. 15

28 Chapter 3 Interference Map Generation 3.1 Radio Environmental Awareness (REA) and Radio Environment Map (REM) Cognitive radios have been introduced as a new paradigm for increasing the efficiency of spectrum utilization, providing more reliable radio services, reducing harmful interference, and helping the cooperativeness of different wireless networks. CRs can autonomously be aware of situations and the radio environment, learn from experience, and adapt by responding to dynamic operational conditions. An essential step before applying any cognitive algorithm is to build a system s radio environmental awareness (REA). To provide CR networks with up to-date global radio environment information, the radio environment map is proposed as an abstraction from real world radio scenarios and as a vehicle of network support [86]. Ideally, REM can offer multi-domain environmental information, such as geographical features, available services, spectral regulations, locations and activities of radios, relevant policies, and experiences. To keep REM information current, updates to the REM database should be made with observations from distributed CR nodes and then distributed throughout the CR network. The REM can also be viewed as an extension to the available resource map (ARM) which has been proposed as a real-time map of all radio activities in the network for cognitive radio applications in unlicensed wide area networks. The idea behind REM is digitizing and indexing radio environment information. The more clearly the radio environment is characterized and modeled, the better the CR can learn from experience and environment. The REM can also incorporate the policy layer, application layer, optimization layer, topology, and network layer information, all of which are important to CR networks. To obtain situation awareness, CRs don t need to conduct sophisticated spectrum sensing algorithms as it maintains or has access to an up-to-date REM through the network support [86]. The REM enables CR to obtain situation awareness in a very efficient way. For example, the REM can inform the CR with what kind of radio networks could be in service at a certain location. Based upon the radio interface specifications stored in the REM database, the CR will know the possible frequency bands and modulation types used by the primary users. The CR can even obtain some prior knowledge of primary users by analyzing the historical REM data and learning from ex- 16

29 Figure 3.1: Spatial interpolation perience. Therefore, CR can conduct primary detection with focused attention instead of spending excessive processing time performing complex spectrum sensing algorithms. This type of approach for primary user detection is very effective in CR networks. Furthermore, REM has the potential to support global cross-layer optimization by enabling CRs to look through various layers: from policy layer, application layer, optimization layer, topology layer, down to the network, MAC, and PHY layers [85]. A typical setting for building REA is to characterize the power profile over a geographical area of interest at a particular time instant. We will refer to it as interference cartography generation. Interference cartography combines measurements performed by different network entities (mobile terminals, base stations, access points) with the geolocation information and applies simple and effective spatial interpolation techniques to achieve a map which indicates the level of interference experienced at each mesh over the area of interest. Using this information, a secondary network can detect the presence of a primary network (or of other secondary networks) and can use spectrum opportunities without causing harmful interference to them. 3.2 RF Cartography Generation Techniques Spatial interpolation [19] is a well-known procedure, commonly used in geographic information systems (GIS). GIS refers to any system manipulating geographical referenced data for capture, storage, analysis and management purposes in many application areas such as agriculture, meteorology, mining, geology, climate and urbanization. Formally, spatial interpolation is a statistical procedure that estimates missing values at unobserved locations within a given area, based on a set of available observations of a random field (see Fig.3.1). One of the most challenging tasks in field reconstruction through interpolation is choosing the most appropriate technique as the validity of each method depends on the type of data being interpolated. The importance of the evaluation of the relative performances of interpolation techniques has been recognized in various fields, including remote sensing 17

30 [25], image reconstruction [41] and wireless sensor networks [69]. However, there is no such relative performance comparison of the interference cartography generation techniques, where the data to be interpolated are the RF field strength values. The RF field strength estimation has been studied in radio propagation prediction literature, where the objective is to predict the field strength values in an area of interest for given emitter location and power. In the context of cognitive radio networks, however, the desire is to estimate the location and power of the primary emitter in addition to estimating the field strength values in any location in the ROI using a limited number of field strength measurements. The spatial interpolation techniques can be classified into two main categories as global and local. In local techniques only the data which fall within the given point s local neighborhood are used for calculating the interpolation values. Global techniques use a weighted sum of all data to do the interpolation. The RF cartography generation techniques studied in this thesis include global interpolation techniques such as thin-plate spline and Kriging interpolation and local interpolation techniques that are based on Delaunay triangulation as they have been proposed as candidates of estimating RF field strength in the literature (refer to Section 2.1). In this chapter, we will give a brief overview of these techniques and study their comparative performance in terms RF field strength estimation and emitter localization for both single and multiple emitter case scenarios Global Interpolation Techniques Thin-plate Spline Interpolation The thin-plate spline interpolation is a global interpolation technique and a special form of radial basis function interpolation. In radial basis function interpolation, a radial basis function is centered on each input sample and the interpolated value f(x), at any given output point x, is calculated as a weighted sum of the contributions from each input point x i : f(x) = N w i φ( x x i ), (3.1) i=1 where N is the number of input points,. is the Euclidean norm, w i is the set of weights and φ(.) is the basis function which is given by φ(r) = r 2 ln(r) (3.2) for thin-plate spline. The values of weights w i can be found by solving the linear system Aw = f (3.3) where A is a matrix of evaluated basis functions for every possible input value and w and f are the column vectors of weights and input values, respectively. This interpolation 18

31 technique has higher computational complexity because of calculation of matrix A and and solving the linear system. Kriging Interpolation Kriging interpolation is also a global interpolation technique and based on empirical semivariograms obtained from the input points, i.e. measurements at sample locations. It consists of the following steps: 1. An empirical semivariogram is formed. Semivariogram is a graph showing the semivariance as a function of separations (distances). It indicates the degree of spatial correlation in measurements at sample locations. The semivariance is defined as: γ(h) = n(h) (f(x) f(x+h)) 2 (3.4) n(h) i=1 where f(x) is the point values at a given location x and f(x+h) is the point value separated from x by the distance vector h and n(h) is the number of paired data at a distance of h. 2. A model is fitted to the empirical semivariogram to form the theoretical semivariogram such as exponential, Gaussian and spherical. 3. The interpolated output value at any given point is determined by using a weighted average of the N input points: f(x) = N w i f(x i ) (3.5) i=1 The weights, w i are calculated by minimizing the kriging variance, which is the difference between the estimated and the actual variance values. Since it minimizes the variance of the output, kriging is often called the best linear unbiased estimator. However, the performance of the interpolation depends on the semivariogram model selection and the parameter N. There is a trade-off between performance and complexity. Larger N values result in better performance at the cost of increased computational complexity. In Fig. 3.2, an empirical semivariogram obtained from RF power measurements is shown. The theoretical semivariogram used to fit the empirical data for this figure is an exponential model. 19

32 200 Emprirical Theoretical γ(h) Lag distance h Figure 3.2: An example of empirical and theoretical semivariograms Local Interpolation Techniques Based on Delaunay Triangulation The Delaunay triangulation and its dual, the Voronoi decomposition are important data structures in computational geometry and have been used in many fields such as cartography and geostatistics. In the following sections the nearest and natural neighbor interpolations are explained by using Voronoi decomposition and the polynomial (linear, quadratic and cubic) interpolations are explained by using Delaunay triangulation for convenience. Delaunay Triangulation For a given set of nodes, Delaunay triangulation of the set is formed by the following steps: 1. Let P be a set of nodes in the plane and C, the convex hull of the nodes. 2. Draw straight lines (not crossing each other) from nodes on the interior to nodes on the boundary of the convex hull (or to each other), until the entire convex hull is divided into a set of polygons, with all the vertices being elements of P. 3. If any of the polygons are not triangles, divide them into triangles by drawing more lines between vertices of the polygons. This will give a triangulation of the set of nodes (see Fig. 3.3). The Delaunay triangulation has the following properties: 20

33 Figure 3.3: Delaunay triangulation of a set of 10 nodes with the circumcircles 1. There is no node in the set that falls in the interior of the circumcircle of any triangle in the triangulation, i.e. the circumcircle of every triangle contains no other nodes. This property is known as Delaunay condition. Triangulation that satisfy the condition is shown in Fig. 3.4 and the one that doesn t satisfy the condition is shown in Fig Every line is also contained within some circle which contains no other nodes. 3. This triangulation maximizes the minimum angle of all the angles of the triangles in the triangulation. 4. This triangulation is unique except for when 4 or more nodes are on the same circle (e.g. the vertices of a rectangle); each of the two possible triangulations that split the quadrangle into two triangles satisfies the Delaunay condition. This can be avoided by choosing in such a way that the other nodes are outside of the circle, thus it is assumed that this case does not occur. Sample Delaunay triangulations of random sets of 20 and 100 points on a 2D plane can be found in Fig. 3.6 and Fig

34 Figure 3.4: Delaunay condition of triangulations (conditions satisfied) Figure 3.5: Delaunay condition of triangulations (conditions not satisfied) 22

35 Figure 3.6: Delaunay triangulations of a random set consisting 20 points Figure 3.7: Delaunay triangulations of a random set consisting 100 points 23

36 A i P i Figure 3.8: Delaunay triangulation and the corresponding Voronoi tessellation of a set of 10 points Voronoi Decomposition The Voronoi decomposition is the partitioning of a plane with points into convex polygons such that each polygon contains exactly one generating point and every point in a given polygoniscloser toitsgenerating pointthantoanyother. Eachnodeinagiven setiscalled Voronoi sites and the decomposition generates a cell for each site. The decomposition can be obtained by connecting the centers of the circumcircles of the Delaunay triangulation. Voronoi decomposition is actually the dual representation of Delaunay triangulation. In Fig. 3.8, an example of the Delaunay triangulation of a set of 10 points is shown with solid lines. The dashed lines show Voronoi decomposition of the same set. Nearest Neighbor Interpolation The nearest neighbor interpolation is straightforward once the Voronoi decomposition of the sensor set is obtained. The RSS value at the given node is assigned to all the points inside the Voronoi cell, i.e. all the points inside a cell take the value of that Voronoi site. For example, all the points inside the cell A 1 in Fig. 3.8 are assigned to the measured value of P 1. Natural Neighbor Interpolation The natural neighbor interpolation is another local interpolation technique based on Voronoi tessellation of a set of given points in the plane. The interpolation consists of the following steps: 24

37 Figure 3.9: Natural neighbor interpolation using Voronoi tessellations 1. The tessellation obtained from the input points serves as a reference for the interpolation. 2. Include the output point (new point) in the input data set and retessellate the resulting set. The new point adds a new Voronoi cell which overlaps with the cells of the reference (see Fig. 3.9) 3. Calculate the interpolated output value at any point x by using a weighted sum of the values at its neighbors x i, i = 1,..,M and M is the number of neighbors of x. The weights w i are given by the ratio of the area of overlap to the total area of the new cell. f(x) = M w i f(x i ) (3.6) i=1 In Fig. 3.9, gray circles represent measurement data locations and the dark circle is the location at which the field is to be estimated. Tessellations before and after adding new data are shown in dashed and solid lines, respectively. The areas between solid and dashed lines determine the weights of the interpolation. Note that the natural-neighbor interpolation technique requires that the points interpolated to be in the convex hull of the measurement locations as the Voronoi cells of outer data points are open-ended polygons with an infinite area. The natural neighbor interpolation is suitable for distributed estimation as it considers information only from the neighbors in the estimation process. 25

38 Polynomial Interpolations The polynomial interpolations we study here are based on Delaunay triangulation. After the triangulation is obtained, each node in the data set is connected to several others by triangle vertices. For a given set of triangle s nodes (f i ), e.g. f i representing a received power level at the corresponding sensor location of (x i,y i ), the interpolated value of any point (x, y) within the triangle is calculated by: f(x,y) = 3 φ i (x,y)f i (3.7) i=1 where φ i (x) is the interpolating basis function that weighs the contributions of the inputs, i.e. readings of the three sensors that are located at the vertices of the Delaunay triangle. For the linear interpolation case the basis function can be replaced by a simple first order polynomial; f(x,y) = c 1 x+c 2 y +c 3 (3.8) The coefficients c = (c 1,c 2,c 3 ) can be found by solving Ac = f where f = (f 1,f 2,f 3 ) T and A is a 3x3 matrix of rows with the form (x i,y i,1), where i is the row number as below. Therefore c = fa 1 x 1 y 1 1 A = x 2 y 2 1 (3.9) x 3 y 3 1 f 1 x 1 y 1 1 c = (c 1,c 2,c 3 ) = f 2 x 2 y 2 1 f 3 x 3 y (3.10) For quadratic and cubic interpolation case, the basis function is a second and third order polynomials respectively. Note that higher order polynomials require larger number of inputs. 3.3 Performance Evaluations of RF Cartography Generation Techniques Network and Signal Model with Assumptions Network Model We consider a cognitive radio network consisting of N s secondary users deployed at known but arbitrary locations in a given geographical area. These nodes act as RF sensors that 26

39 measure the received signal strength due to the primary transmitter(s) at a given frequency and bandwidth. The secondary radios report their power measurements and location information to a fusion center through a common channel. The fusion center processes this information to estimate the interference level over the entire area so that a map of the RF field strength of the considered region can be constructed. We assume that individual sensor measurements are perfect and there is no information loss in the delivery of sensor measurements to the central node. For simplicity, we first define the signal model for the single emitter case scenario. Additionally, the signal model for the multiple emitter case scenario is given in Section The ROI is assumed to be 1 km square area with a certain constraint on the emitter and sensor geometries. For the single emitter case scenario, all sensors are assumed to be at least 50 m away from the emitter, a constraint required to guarantee that the log-distance propagation model yields realistic results [28]. Signal Model We assume that all transmissions are omnidirectional and the signal propagation is governed by a distance dependent path loss model such that the noise-free received power at the jth receiver, j = 1, 2,...,N s from the primary transmitter is given by s j = βp t (d j ) α (3.11) where P t is the emitter power, β is a constant that reflects the carrier frequency and antenna properties given by β = ( λ 4π )2, λ is the wavelength, α is the path-loss exponent and d j is the distance between sensor j and the primary emitter located at θ = (T x,t y ). We assume that each sensor experiences log-normal shadowing. If the fast fading effects are sufficiently averaged then the resulting unknown measured power from the emitter to the jth sensor is given by r j = s j 10 W 10 (3.12) where W N(0,σ 2 ) is the gain/loss in db due to shadowing and σ is called the shadowing spread. The received power at a distance from the emitter is a log-normal random variable with mean s j and variance σ 2 and its pdf is given by where ǫ = ln p(r j ;s j,σ) = 1 σ 2πǫr j exp ( (10log 10r j 10log 10 s j ) 2 ) 2σ 2 (3.13) Performance Evaluations for RF Field Strength Estimation In this section we compare the RF field strength estimation performances of the interpolation techniques commonly used in the generation of interference cartographs. Obtaining 27

40 field strength estimates is important, specifically in setting the acceptable transmit power for secondary users. We consider a simulated CR network and a propagation model as described in Section Sensors are randomly deployed at locations on a planar square of length 1 km with the given emitter and sensor geometry restriction above. A path-loss exponent of 3 is assumed. The shadowing values depend on the terrain structure and they show a spatial correlation as terrain structures do not show fast spatial changes in real environments. Random shadow fading is generated from log-normal distribution with given spread at points separated by the decorrelation distance d corr, which is the distance at which shadow fading becomes independent. The correlated shadow fading values in between independent locations are then computed by applying spline interpolation [24]. A sample figure that shows the shadow loss of a 1km by 1km area, generated at a shadowing spread of 6 db with a decorrelation distance of 50 m is illustrated in Fig In this figure, the correlated shadow values between indepedent points at which random shadow fading is generated, are calculated for each 1m by 1m grid points and the calcaluted values are transformed into a colored image. Simulated received RF strength values at given sensor locations are used to obtain the interpolant function for any given interpolation method. After the area is divided into grids, the RF field strength at each tile center is interpolated by using the given interpolant function. Hence an interference map is generated over the area of interest. For our simulations, the grid size is taken as 100 m by 100 m. It should be noted that, the grid is used only to determine the points at which the the RF strength values will be estimated and not utilized in calculating the interpolants. Hence a quantization error due to grid size is not a concern for our estimations with any of the interpolation techniques unlike the grid search method described in Section Assumingthatonlyoneprimaryuserisactivewith1Wemitterpoweratanygiventime, performance evaluations of the interpolation techniques are studied in two categories; first performances of global methods are compared with a local method s performance, second the performances of local methods based on Delaunay triangulation are compared within each other. Note that, the knowledge of the propagation model and/or any parameter related with the environment such as path-loss exponent, shadowing spread or emitter power are not taken into account for any of the interpolation methods since these techniques do not use this knowledge that may be available and they require only the measured received power values at sensors and their locations to make estimations. Global and Local Interpolation Techniques: Sensor measurements are simulated based on the correlated shadowing model and given sensor and primary emitter locations. The simulations are performed for networks with 100 nodes. A number of sensors are selected randomly among all sensors and their measurements are used to calculate the interpolant functions. The RF strength values at the remaining nodes are then estimated based on the interpolants using a local that is natural neighbor and global interpolation techniques that are thin-plate spline and kriging interpolation techniques. Note that output sites that are outside the convex hull of the input sites are excluded in performance evaluations, as these techniques cannot extrapolate outside 28

41 y [m] Shadow loss [db] x [m] Figure 3.10: Example of two-dimensional shadow fading map with spline interpolated (correlated) shadow fading variables for σ = 6 db, α = 3, L = 1 km, d corr = 50 m the convex hull of the input sites. An example simulation of sensor placements can be seen in Fig In this figure 20 sensors (input sites) shown with red circles are used to estimate the location of remaining 80 sensors (output sites) shown with blue crosses. The large red polygon that surrounds the input sites, shows the convex hull in which the output sites can be interpolated as the Voronoi cells of outer data points are open-ended polygons with an infinite area. The Voroni tesselation of input sites and their convex hull is also illustrated in Fig The difference between the estimated and simulated RF strength values in db determine the estimation error, E. We define the efficiency, η is defined as the probability that the RMS value of the RF power estimation error, E rms is less than the db spread of the shadowing, σ; η = Pr(E rms < σ) (3.14) Sensor measurements simulated at 20 nodes are used to estimate the RF strength values at the remaining 80 nodes using the three interpolation techniques. The efficiency is calculated for Monte Carlo simulations for db spread values ranging from 1 to 10 db at 1 db intervals and decorrelation distances of 50 to 500 m at 50 m intervals. Fig shows the efficiency results averaged over the shadowing spread values as a function of decorrelation distance. As can be seen from the figure, average efficiency of all interpolation techniques improves with increased shadowing decorrelation. The performance difference between the interpolation methods is negligible when the decorrelation distance is higher than 250m. 29

42 y [m] x [m] Figure 3.11: A simulation example: RF field strength estimation at output sites using input sites (N s = 100,N selected = 20) y [m] x [m] Figure 3.12: Voronoi tesselation and convex hull of input sites (N s = 100,N selected = 20) 30

43 Average efficiency Natural neighbour Kriging Thin plate spline Decorrelation distance(m) Figure 3.13: Average efficiency of the RF field strength estimates in correlated shadowing for estimating the RF power values at 80 nodes by using the RF power values at 20 nodes (N t = 1, N s = 100, N selected = 20, α = 3, P t = 1 W, L = 1 km, grid size = 100m by 100m) The effect of number of interpolation points to the efficiency of the RF power estimates is also evaluated by varying the number of nodes. In the simulations the number ofnodes is rangingfrom10to40. Shadowing spreadanddecorrelationdistancearesetto6dband250 m respectively. The calculated efficiencies are shown in Fig As seen from the figure, the efficiency of the thin-plate spline interpolation is better than the other techniques, however the performance gap becomes negligible as the number of sensors increase. The thin-plate spline interpolation outperforms the other two interpolation techniques in terms of RF field strength estimation. Note that the kriging and thin-plate spline methods are both global interpolation techniques which require a central node to process the information from individual nodes, but natural neighbor interpolation is a local technique. On the other hand, local interpolation techniques such as natural neighbor interpolation, require information only from the neighboring nodes; thus a central fusion node is not required. This characteristic may provide desirable features such as robustness and better adaptation to local changes, for example due to mobility in CRNs. For this reason local interpolation techniques are suitable for distributed estimation. Particularly, in the context of ad hoc networks where the nodes can form into a network without an infrastructure, it is more efficient to perform local computations rather than implementing network wide updates. 31

44 Efficiency Natural neighbour Kriging Thin plate spline Number of sensors Figure 3.14: Efficiency of the RF field strength estimates in correlated shadowing as a function of number of sensors (σ = 6 db, d corr = 250 m, N t = 1, N s = 100, N selected = 10 40, α = 3, P t = 1 W, L = 1 km, grid size = 100 m by 100 m) Delaunay Based Interpolation Techniques: In this section we compare the performances of the local interpolation techniques based on Delaunay triangulation in generating interference cartographs. These are nearest neighbor, natural neighbor, linear, quadratic and cubic interpolation techniques. Here we consider an ad hoc network consisting of secondary users (nodes) deployed at known but arbitrary locations in a given geographical area as described in Section We assume that the nodes discovering their one-hop neighbors are triangulated for networking tasks such as routing [12] or topology control [40], and our aim is to utilize the existing triangulation for interference map generation in order to avoid complex computations while generating the interference map. Each node measures the local RSS due to the primary emitter at a given frequency and sends the information to its one-hop neighbors. The nodes process RSS measurements using the Delaunay triangulation to estimate the field strength values by local interpolation techniques. Sensor measurements simulated at 20 randomly selected nodes are used to find the interpolant functions and then estimation of the RF strength values at the remaining 80 nodes are done based on the calculated interpolants using the five interpolation techniques. The efficiency was calculated over Monte Carlo simulations for db spread value of 6 db and decorrelation distances of 50 to 250 m at 50 m intervals. Fig shows the field strength estimation efficiency results averaged over the shadowing spread values as a function of decorrelation distance. As seen from this figure, average efficiency of all in- 32

45 Natural Nearest Linear Cubic Quadratic 0.5 Efficiency Decorrelation distance Figure 3.15: Efficiency of RSS estimates in correlated shadowing as a function of decorrelation distance (N t = 1, N s = 100, N selected = 20, α = 3, σ = 6 db, P t = 1 W, L = 1 km, grid size = 100m by 100m) Efficiency Natural Nearest Linear Cubic Quadratic Number of sensors Figure 3.16: Efficiency of RSS estimates in correlated shadowing as a function of number of sensors(σ = 6 db, d corr = 250 m, N t = 1, N s = 100, N selected = 10 40, α = 3, P t = 1 W, L = 1 km, grid size = 100 m by 100 m) 33

46 terpolation techniques improves with increased shadowing decorrelation. The performance differences between the interpolation methods are negligible except for the nearest neighbor interpolation that performs poorly. The effect of number of interpolation points to the efficiency of the RF power estimates is also shown in Fig as simulations are repeated by varying the number of nodes ranging from 10 to 40. In all simulations, shadowing spread and decorrelation distance are set to 6 db and 250 m respectively. As seen from the figure, the efficiency of nearest neighbor interpolation is worse than any of the other techniques that have comparable performances. As a result, among the five considered techniques above, nearest neighbor interpolation is the least complex method, however its performance is the worst. Since all the other techniques based on Delaunay triangulation have comparable performances, linear interpolation would be better than the other RSS interpolation techniques to use because of its less complexity. 3.4 Emitter Localization Using the Generated Radio Environment Map An interference cartograph provides field strength values at any desired location in the area and emitter location can be estimated simply by selecting the peak location of the field. Even though this estimator has lower accuracy compared to the ML estimator, the achieved accuracy level may be enough for applications where a high level of accuracy is not required. Besides, rough estimates obtained from the interference cartograph can be used to initialize more accurate localization algorithms to reduce their computational complexity or to improve estimation accuracy. (Refer to Section 5.2) In our simulations in Section 3.3.2, the emitter s location is estimated at the center of the tile that contains the peak value of the interpolated RF field strength in the given area. Performance evaluations of the interpolation techniques are also presented in terms of primary emitter localization accuracy. We define localization error as the difference between estimated and the actual transmitter location and calculate the root mean squared error (RMSE) of localization for each interpolation technique by averaging the simulations Single Emitter Case Performance Comparison of Global and Local Interpolation Techniques: Assuming that one primary emitter is active with 1 W emitter power at any given time, simulations have been run for db spread values ranging from 1 to 10 db at 1 db intervals for independent shadow fading case and the number of sensors ranging from 5 to 30. Fig shows the RMSE performance versus shadowing spread for networks of 10 sensors and Fig displays the RMSE performance versus number of sensors for shadowing spread of 6 db. As seen from these figures, both kriging and natural-neighbor interpolations perform 34

47 Natural Krigging Thin-plate spline ML RMSE [m] Shadown spread [db]) Figure 3.17: Localization errors of cartography generation techniques at different shadowing spread values with uncorrelated shadowing (N t = 1, N s = 10, α = 3, P t = 1 W, L = 1 km, grid size for ML = 10 m by 10 m) similarly when the channel and/or measurement uncertainty is lower. When the channel and/or measurement uncertainty increase, natural-neighbor interpolation technique outperforms the kriging in terms of primary emitter localization performance. For a baseline comparison for emitter localization, the performance of the ML estimator based on grid search is also included in Fig and The ML estimator is derived based on the assumption that the received signal strength values are independently distributed, each having a log-normal density. The likelihood of receiving sensor measurements is calculated for each tile on a regular grid and the center of the tile that gives the maximum likelihood value is selected as an estimate of the emitter location. To implement the ML method, we divided the one square kilometer simulation region into 10 m by 10 m square tiles for which the likelihood values are calculated. As seen from the figures, the ML estimator based on grid search provides more accurate primary emitter location estimates than the three interpolation techniques. It should be noted that, the RMSE error of ML estimate at zero shadow spread is 5 m. However, it requires the knowledge of the propagation model and the path-loss exponent. Furthermore, it is computationally more complex as it requires the calculation of the likelihood function for each tile in the grid. 35

48 Natural neighbor Kriging Thin-plate spline ML estimate RMSE [m] Number of sensors Figure 3.18: Localization errors of cartography generation techniques as a function of number of sensors with uncorrelated shadowing (σ = 6 db, N t = 1, N s = 100, α = 3, P t = 1 W, L = 1 km, grid size for ML = 10 m by 10 m) RMSE [m] Natural Nearest Linear Cubic Quadratic Shadowing spread (db) Figure 3.19: Comparison of localization errors at different shadowing spread values for 10 sensors (N t = 1, N s = 10, α = 3, P t = 1 W, L = 1 km, grid size = 100 m by 100 m) 36

49 Natural Nearest Linear Cubic Quadratic 400 RMSE [m] Number of sensors Figure 3.20: Comparison of localization errors as a function of number of sensors (σ = 6 db, N t = 1, α = 3, P t = 1 W, L = 1 km, grid size = 100 m by 100 m) Performance Comparison of Delaunay Based Interpolation Techniques: Simulations have been run for db spread values ranging from 1 to 12 db at 1 db intervals and the number of sensors ranging from 5 to 40. Figure 3.19 displays the RMSE performance against shadowing spread for networks of 10 sensors and Fig shows the RMSE performance versus number of sensors in a channel with shadowing spread of 6 db. As seen from the figures, the localization performances of all interpolation techniques degrade as the shadowing spread increases. Increasing the number of sensors improves the localization performance. In these figures, the RMSE values of natural neighbor and linear interpolation are the same, therefore their curves overlap. Nearest neighbor interpolation has the worst performance among the others. All the other techniques have comparable performances Multiple Emitter Case In the above simulations, we have studied the RF cartography generation techniques with the assumption that one active primary emitter exists in the network. When there are more than one primary emitters, all transmitting at the same time, the localization becomes quite challenging since there will be more than one peak of the RF field strength values to find in the reconstructed interference map. In this section we study the RF cartography generation techniques that are based on Delaunay triangulation for multiple emitter case. We first reconstruct the interference map from the simulated RF field strength measurements at 37

50 sensor locations and estimate the locations of emitters from the reconstructed map. Note that we assume all active primary transmitters are randomly located and all have equal transmit power value of 1 W with a certain constraint on the emitter and sensor geometries. The constraints and the selection of the value of other network parameters are justified in Section considering a microcellular network with a cluster size of 3 and cell size of 100 m in detail. The first constraint is that the emitters are assumed to be seperated by at least 300 m, reflecting the physical reality that of primary transmitters using the same frequency band would interfere if they were too close together. The second constraint is, all sensors are assumed to be at least twice the reference distance (assumed to be 1 m for our simulations) from all emitters. This is a constraint required to guarantee that the logdistance propagation model yields realistic results [28] and taken as 10 m. The minimum distance between sensors is also assumed to be 10 m. The number of active emitters at a time is also assumed to be known which sets the number of peaks to be found in the reconstructed map. When a peak is found, no other peak with a position within a 300 m by 300 m square area centered at the first peak is detected. Peaks are found sequentially; for example, after the highest peak has been found, the second is found at the largest value in the estimated RF strength values excepting the first found peak. A detailed explanation of finding the number of the peaks and the location of the peaks of a 2D array can be found in [68]. In our simulations, a network of 100 randomly located sensors with a path-loss exponent of 3 in a 1 km by 1 km area is considered. Simulated RF field strength values at the sensors are used to estimate the field strength values on a 10 m by 10 m grid using a particular interpolation technique. We have run the simulations for 3, 4 or 5 emitters being active in the network under different shadowing spread values ranging from 0 to 6 db with nearest neighbor, natural neighbor and linear interpolation methods. Examples of simulation results are illustrated in Fig through Fig Samples of the constructed cartographs with nearest interpolation method for 5 emitters under 4 db of shadowing spread is shown in Fig Similarly, maps constructed with natural neighbor and linear interpolation methods for 5 emitters with 4dB shadowing spread values are shown in Fig and Fig respectively. In these figures, the interpolated RF field strength measurements are transformed into a colored image where the lowest field strength is mappedintoablueandthehigheststrengthismappedintoayellowcolor. Thecolorbaron therightsideshowsthepowerlevelsbetween-35and-70dbm. Theredcirclesrepresent the actual locations of emitters that are randomly generated and the blue rectangles represent their estimated locations. Ideally, the estimated emitter locations would lie on the yellow colored regions as the estimates are done at the center of the tile that contains the peak value of the interpolated RF field strength in the given area. Although the estimated emitter locations were quite close to the actual locations on the illustrated simulation results, there are some occasions where peaks might be missed and estimates might fall further away from the peaks yielding in high location estimation errors. An example is illustrated in Fig for estimating the locations of 5 emitters by using the measurements of 30 sensors placed randomly in the area with natural neighbor interpolation. Simulated field strength values at the sensors are used to estimate the field strength values on a 10 m by 10 m grid using natural interpolation technique for a db 38

51 y [m] Nearest neighbor Field strength [dbm] x [m] Figure 3.21: Nearest neighbor interpolation with multiple emitters (N t = 5, N s = 100, σ = 4 db, α = 3, P t = 1 W, L = 1 km, grid size = 10 m by 10 m, red circles and blue rectangles are actual and estimated emitter locations, respectively) 1000 Natural neighbor y [m] Field strength [dbm] x [m] Figure 3.22: Natural neighbor interpolation with multiple emitters (N t = 5, N s = 100, σ = 4 db, α = 3, P t = 1 W, L = 1 km, grid size = 10 m by 10 m, red circles and blue rectangles are actual and estimated emitter locations, respectively) 39

52 1000 Linear y [m] Field strength [dbm] x [m] Figure 3.23: Linear interpolation with multiple emitters (N t = 5, N s = 100, σ = 4 db, α = 3, P t = 1 W, L = 1 km, grid size = 10 m by 10 m, red circles and blue rectangles are actual and estimated emitter locations, respectively) spread of 4 and a path-loss exponent of 3 within a 1 km by 1 km area. In the figure, the sensor locations are shown with red crosses. Idealy, we expect to see five peaks (estimates) near the actual emitter locations. However, findings for this particular example is quite different. As can be seen from the figure, the four estimates are near the actual locations, but the estimate of the emitter which is at the top right corner with the coordinates (950, 1000) is found to be further away from its actual location causing an high estimation error. This is due to the lack of sensors around that emitter whereas the other four emitters are surrounded by enough number of sensors. At this point, we can conclude that not only the number of sensors affects the estimation accuracy, but also the emitter-sensor geometry has an impact on the estimation performance. We discuss this issue in detail in the next chapter in Section A comparison of RF field strength estimation performances of the interpolation techniques for multiple emitter cartography generation is also studied. We consider a network of 200 randomly located sensors with a path-loss exponent of 2.5 in a 1 km by 1 km area. In order to analyze the error in estimating the RF field strength in an area of interest, 100 received power measurements at the sensors (input sites) that are randomly selected among the simulated measurements are used to obtain the Delaunay triangulation which forms the interpolant function. The field strength values at the remaining sensor locations (output sites) are then estimated based on the calculated interpolants using nearest neighbour, natural neighbor and linear interpolation techniques based on Delaunay triangulations. Note that the output sites that are outside the convex hull of the input sites are 40

53 Natural neighbor y [m] Field strength [dbm] x [m] Figure 3.24: Natural neighbor interpolation with high estimation error (N t = 5, N s = 30, σ = 4 db, α = 3, P t = 1 W, L = 1 km, grid size= 10 m by 10 m, red crosses are sensor locations, red circles and blue rectangles are actual and estimated emitter locations, respectively) excluded in the performance evaluations, as these techniques cannot extrapolate outside the convex hull of the input sites. An example of a Delaunay triangulation formed with 100 randomly selected sensors among 200 randomly located ones in a square region of 1 km by 1 km can be seen in Fig In this figure, the power levels of sensors whose locations are shown with blue circles are used to estimate the power levels of the sensors with the locations shown in red. Four of the sensors are placed at the corners of the square region such that all sensors lie within the convex hull of the triangulation. Therefore each and every sensor measurement is included in performance evaluations. Estimation errors are calculated as the differences between the estimated and simulated field strength values in db. The simulations are performed with multiple primary emitters of 1 to 5. In order to evaluate the effect of shadow fading, the probability of RF field strength estimation error being less than 5dB is plotted as a function of db spread values ranging from 0 to 12 db with different number of transmitters in the network for the nearest neighbor, natural neighbour and linear interpolation methods in Fig. 3.26, Fig and Fig respectively. The probability of the estimation error being less than 10 db for the three interpolation methods is also given in Fig. 3.27, Fig and Fig respectively. As can be seen from these figures, the estimation error increases as the shadowing spread in db increases, as expected. But in terms of number of transmitters there isn t a certain proportion for the probability of error against db spread, i.e. the effect of the number of transmitters to the power estimates depends on the shadowing spread. For shadowing spread of 3 db or less, the estimation error increases with the increased 41

54 Figure 3.25: A Delaunay triangulation formed with 100 sensors to interpolate the power levels at the remaining sensors (N s = 200, N seleceted = 100) number of transmitters in all methods. However for shadowing values bigger than 3 db, the estimation error decreases with the increased number of transmitters which can be interpreted as the increase in the number of transmitters helps reducing the shadowing effects. A comparison of field strength estimation error at different shadowing spread values for three transmitters is plotted for the three interpolation methods for 5 dbm and 10 dbm in Fig and Fig respectively. As seen from the figures, the estimation error increases for bigger values of shadowing spread. The worst performance among the three interpolation methods belongs to the nearest neighbour where natural neighbor and linear interpolation methods provide comparable performance levels. In terms of computational complexity, linear interpolation can be preferred for further analysis simulations have been run for a number of selected sensors ranging from 50 to 150 using linear interpolation to estimate the RF field strength value at the rest of 200 sensors, as linear interpolation is advantageous due to its higher performance and lower computational complexity. The probability that the RF field strength estimation error is less than 5 db and 10 db is plotted as a function of the number of sensors for shadowing spread value of 6 db for 1 to 5 transmitters in Fig and Fig respectively. As seen from the figures, the RF field estimation error decreases with the increased number of sensors. For a shadowing spread of 6 db, the estimation error decreases with the increased number of transmitters as the increase in the number of emitters helps reducing the shadowing effect for higher shadowing values. The cdf of the RF field strength estimation error 42

55 Tx 2 Tx 3 Tx 4 Tx 5 Tx Prob[e<5dB] Shadow fading spread (db) Figure 3.26: Prob. of RSS estimation error being less than 5 db as a function of shadowing spread for nearest neighbor interpolation for multiple emitters (N t = 1-5, N s = 200, N selected = 100, α = 2.5, P t = 1 W, L = 1 km, grid size = 10 m by 10 m) Tx 2 Tx 3 Tx 4 Tx 5 Tx 0.8 Prob[e<10dB] Shadow fading spread (db) Figure3.27: Prob. ofrssestimationerrorbeinglessthan10dbasafunctionofshadowing spread for nearest neighbor interpolation for multiple emitters (N t = 1-5, N s = 200, N selected = 100, α = 2.5, P t = 1 W, L = 1 km, grid size = 10 m by 10 m) 43

56 Tx 2 Tx 3 Tx 4 Tx 5 Tx Prob[e<5dB] Shadow fading spread (db) Figure 3.28: Prob. of RSS estimation error being less than 5 db as a function of shadowing spread for natural neighbor interpolation for multiple emitters (N t = 1-5, N s = 200, N selected = 100, α = 2.5, P t = 1 W, L = 1 km, grid size = 10 m by 10 m) Tx 2 Tx 3 Tx 4 Tx 5 Tx Prob[e<10dB] Shadow fading spread (db) Figure3.29: Prob. ofrssestimationerrorbeinglessthan10dbasafunctionofshadowing spread for natural neighbor interpolation for multiple emitters (N t = 1-5, N s = 200, N selected = 100, α = 2.5, P t = 1 W, L = 1 km, grid size = 10 m by 10 m) 44

57 Tx 2 Tx 3 Tx 4 Tx 5 Tx Prob[e<5dB] Shadow fading spread (db) Figure 3.30: Prob. of RSS estimation error being less than 5 db as a function of shadowing spread for linear interpolation for multiple emitters (N t = 1-5, N s = 200, N selected = 100, α = 2.5, P t = 1 W, L = 1 km, grid size = 10 m by 10 m) Tx 2 Tx 3 Tx 4 Tx 5 Tx 0.8 Prob[e<10dB] Shadow fading spread (db) Figure3.31: Prob. ofrssestimationerrorbeinglessthan10dbasafunctionofshadowing spread for natural neighbor interpolation for multiple emitters (N t = 1-5, N s = 200, N selected = 100, α = 2.5, P t = 1 W, L = 1 km, grid size = 10 m by 10 m) 45

58 1 0.9 Nearest Natural Linear 0.8 Prob[e<5dB] Shadow fading spread (db) Figure 3.32: Comparison of RSS estimation error probability for 5 db for nearest n., natural n. and linear interpolation methods at different shadowing spread values (N t = 3, N s = 200, N selected = 100, α = 2.5, P t = 1 W, L = 1 km, grid size= 10 m by 10 m) Nearest Natural Linear 0.85 Prob[e<10dB] Shadow fading spread (db) Figure 3.33: Comparison of RSS estimation error probability for 10 db for nearest n., natural n. and linear interpolation methods at different shadowing spread values (N t = 3, N s = 200, N selected = 100, α = 2.5, P t = 1 W, L = 1 km, grid size= 10 m by 10 m) 46

59 Prob[e<5dB] Tx 2 Tx 3 Tx 4 Tx 5 Tx Number of sensors Figure 3.34: Prob. of RSS estimation error being less than 5 db as a function of number of sensors for linear interpolation for multiple emitters (N t = 1 5, N s = 200, N selected = , α = 2.5, σ = 6 db, P t = 1 W, L = 1 km, grid size= 10 m by 10 m) of linear interpolation method is also plotted for 6 db shadowing spread and 100 sensors in Fig to provide further insights. A comparison of received power estimation error for 5 db and 10 db as a function of number of sensors for three transmitters under 6 db shadowing spread is plotted for the three interpolation methods in Fig and Fig respectively. As seen from the figure, the estimation error decreases with the increased number of sensors. The nearest neighbour interpolation technique has the worst performance among the three interpolation methods and natural neighbor and linear interpolation methods provide comparable performance levels. Between natural neighbor and linear interpolation, linear interpolation has lower computational complexity than natural neighbor interpolation. 47

60 Prob[e<10dB] Tx 2 Tx 3 Tx 4 Tx 5 Tx Number of sensors Figure 3.35: Prob. of RSS estimation error being less than 10 db as a function of number of sensors for linear interpolation for multiple emitters (N t = 1-5, N s = 200, N selected = , α = 2.5, σ = 6 db, P t = 1 W, L = 1 km, grid size= 10 m by 10 m) CDF of power estimation error Tx 2 Tx 3 Tx 4 Tx 5 Tx Power estimation error (db) Figure 3.36: CDF of RSS estimation error for linear interpolation based on Delaunay triangulation (N s = 200, N selected = 100, α = 2.5, σ = 6 db, P t = 1 W, L = 1 km, grid size = 10 m by 10 m) 48

61 Nearest Natural Linear 0.6 Prob[e<5dB] Number of sensors Figure 3.37: Comparison of RSS estimation error probability for 5 db as a function of number of sensors for three transmitters (N t = 3, N s = 200, N selected = , α = 2.5, σ = 6 db, P t = 1 W, L = 1 km, grid size= 10 m by 10 m) Nearest Natural Linear 0.89 Prob[e<10dB] Number of sensors Figure 3.38: Comparison of RSS estimation error probability for 10 db as a function of number of sensors for three transmitters (N t = 3, N s = 200, N selected = , α = 2.5, σ = 6 db, P t = 1 W, L = 1 km, grid size= 10 m by 10 m) 49

62 Chapter 4 Improving Emitter Localization Accuracy As shown in the previous section (see Fig and 3.18), the location estimates obtained from interference maps are not very accurate. Specifically the interference map introduces estimation errors even when there is no uncertainty due to shadowing because of using limited sensor measurements to predict the RF strength values at grid locations throughout the entire region of interest (ROI). Interpolation is a suboptimal estimator because it does not use any knowledge that may be available regarding the propagation model and/or any parameter related with the environment such as path-loss exponent. This type of localization can only give us rough estimates of the emitter location. However this information can be further utilized to initialize more accurate localization algorithms. A detailed study on initialization of an emitter localization algorithms by the information extracted from a REM is presented in Chapter 5. As a result, a need to look for more accurate algorithms arises for emitter localization problem in the thesis. As mentioned in the literature survey, there is an extensive work on efficient algorithms for accurate localization. Among RSS-based emitter localization techniques, MLE method is the most attractive approach as it is an unbiased linear estimator and gives estimates that are close to the Cramer Rao lower bound (CRLB) [65]. We will first study the MLE technique for improving the localization accuracy and analyze its performance in comparison with the bound. However, it is important to analyze CRLB and investigate the effects of model parameters on the bound to understand inter-relation between the parameters and the estimation accuracy. A theoretical background and some simulation results for CRLB is presented in the next section. For simplicity, before we delve into more accurate localization algorithms for multiple emitter localization, we first study these algorithms and evaluate their performances for single emitter localization problem. 50

63 4.1 Single Emitter Localization Cramer Rao Lower Bound for Emitter Localization Error The accuracy of the RSS-based emitter location estimators can be best understood by analyzing the CRLB of the estimates which provide information on the optimal performance achievable for unbiased estimators. The CRLB is a bound on the covariance matrix of Fisher information matrix of unbiased estimates of the model parameters such as emitter locations, transmission power and/or path-loss exponent. In particular, the diagonal elements of the inverse of Fisher information matrix determine the CRLB bound on the variance of the unbiased estimates of the corresponding parameter. The components of the Fisher information matrix (FIM), J i,j are defined by; { } 2 ln( L) J i,j = E z i z j (4.1) where ln( L) is the log-likelihood function, E denotes the statistical expectation operator, z is the unknown parameter to be estimated [45]. Here i and j represent the row and column index of FIM. By computing the CRLB, we determine the lowest possible estimation variance we can achieve, regardless of which algorithm or method is used in the estimation process. A variance equal to the CRLB is not necessarily achievable, but in practice it is usually possible to get very close to the bound with some estimators. In this section, we compute the CRLB for the emitter location parameter only and use the bound to analyze the performances of different location estimators under different simulation conditions. There are many papers where the accuracy of RSS-based location estimators has been analyzed through the explicit computation of the Fisher information matrix for unbiased location estimators. In [80], a CRLB was derived for estimates of the 2D coordinates and omni-directional transmit power of the source using RSS measurements. Similarly in [55], the CRLB and a maximum likelihood (ML) estimator were derived under the same conditions for self-localization of a network of sensors, in which few of the sensors in the network were reference nodes at known locations. In [78], ML emitter location estimators are derived for both RSS and RSS difference (RRSD) information under the assumption of a log-normal path loss model and these two ML estimators are shown to be identical. To provide a reference standard, the Fisher information matrix and the CRLB for the root mean square (RMS) of emitter localization error which is defined as average miss distance in [78], have also been derived for unbiased RSSD-based emitter estimators. In [79], CRLB of the RMS of localization error for RSS-based geolocation is derived under the same assumptions. It is shown that the bound derived in this paper is affected by the number of sensors, the path-loss exponent, the shadowing spread and a particular parameter G, which depends only on the relative positions of the emitter and receivers and is called the geometric dilution of precision (GDOP) for the given emitter-sensor geometry. 51

64 Similarly in [42], a CRLB derivation for location under the log normal model is reviewed as well. To provide a good practical understanding of the CRLB, we calculate the bound values forshadowing spreadvaluesrangingfrom1dbto12dbby usingequations thatarederived in [79] and [42] for a given emitter-sensor geometry. The CRLB expressions used in these two papers are in fact identical, except that the one in [42] is just the matrix representation of the form given in [79] which is given below. The CRLB for the RMS of emitter localization error for RSS-based geo-localization under log-normal shadowing is given by [79]; where CRLB(κ) = ln10 10 ) κ = E (( T x T x ) 2 +( T y T y ) 2 σ α G (4.2) N s (4.3) Recall N s is the number of sensors, T x, T y, T x, and T y are the actual and estimated emitter locations in x and y directions, respectively, σ is the shadowing spread and α is the pathloss exponent. Here E(.) represents the statistical expectation operator and the parameter G is given by; G = τ 2 +ρ 2 (τ) 2 (ρ) 2 (4.4) 1 τ ρ τ τ 2 (τρ) ρ (τρ) ρ 2 Here, τ and ρ are the averages of τ j and ρ j, τ 2 and ρ 2 are the averages of τj 2 and ρ 2 j, τ j = cos(φ j) d j, ρ j = sin(φ j) d j, τρ is the average of τ j ρ j, and φ j is the angle between the vector emanating from the jth sensor to the emitter and the positive direction of the x-axis of the reference coordinate system and d j the distance between the emitter and jth sensor. A sample figure that shows the distance and angle for an emitter located at [500, 500] and a sensor located at [650, 850] coordinates is depicted in Fig Note that the shadowing spread is assumed to be constant for each emitter-sensor pair. As seen from (4.4), the CRLB depends not only on the shadowing spread, path-loss exponent, the number of sensors, but the emitter-sensor geometry as well which means each unique emitter-sensor placement results in a different CRLB. However, the value of the emitter power has no effect on the bound. CRLB Calculations for Uniform Geometries To gain more insights of the CRLB, we plot CRLB bound values for random and uniform sensor geometries with three different scenarios. In all three scenarios, an independent log-normal shadowing of given db spread value and a log-distance path-loss model with a 52

65 jth sensor d j 600 y [m] emitter φ j x [m] Figure 4.1: Distance and angle for parameter G in calculating CRLB fixed path-loss exponent of 4 is assumed. In these scenarios, emitter and sensors are placed in a 1 km by 1 km square ROI. Scenario 1: In this scenario, 100 sensors (denoted by black dots) are placed in a square lattice with uniform spacing between them and the emitter (denoted by red cross), is placed at the center of the ROI, as shown the Fig The CRLB for this case is calculated for db spread values of 1 to 12 db and plotted in Fig. 4.4 as the red line. CRLB Calculations for Random Geometries Scenario 2: In this scenario, the emitter and the sensors are placed randomly with uniform distribution in the ROI. One random realization of this placement is depicted in Fig The mean, minimum and maximum CRLB values calculated over 1000 different realizations are plotted in Fig As seen from the Fig. 4.4, uniform emitter sensor geometry generally results in smaller CRLB values when compared to the random placement geometries. However, there are some random placements that give smaller bound values than the uniform placement. Scenario 3: In this case, 100 sensors are placed in a square lattice with uniform spacing between them as in Scenario 1 over 1 km by 1 km area. The CRLB is calculated for each location where the emitter is assumed to be placed at the center of 10 m by 10 m grid locations, creating a 100 pixel by 100 pixel image, as shown in Fig In this image, CRLB values are color coded as shown in the color bar. The bound values from smaller to large are represented by blue to red color respectively. As seen from this figure, CRLB 53

66 y [m] x [m] Figure 4.2: Uniform sensor geometry (N t = 1, N s =100, L = 1 km) y [m] x [m] Figure 4.3: Random emitter sensor geometry (N t = 1, N s =100, L = 1 km) 54

67 Uniform Mean Min Max 120 CRLB [m] Shadow spread (db) Figure 4.4: CRLB for the RMS of location error for uniform (see Fig. 4.2) and random emitter-sensor geometry (see Fig. 4.3) (N t = 1, N s = 100, L = 1 km, α = 4) values are smaller when the emitter is placed inside the region covered by the sensors, yielding more accurate location estimations. When an emitter is placed at the center of a grid where a sensor is placed, the value of the bound increases, yielding in poor location estimations. As can be seen from Fig. 4.4 and Fig. 4.5, smaller CRLB values are achieved when sensors are placed around an emitter. In other words, bound values increase as the sensors are placed further away from the emitter. In accordance with this result, another result obtained in [20] showed that the optimal sensor placement is achieved when the sensors are distributed with uniform spacing around the emitter in a circle Single Emitter Localization Using MLE(Grid Search) Method The emitter location and received power parameters in (3.11) are considered to be random variables. Assuming that the RSS values are independently distributed, each having a log-normal shadowing, the likelihood of observing all N s sensor outputs given the signal model in (3.11) and (3.12) is written as L = p(r M,θ) = N s j=1 2 1 σ e ( 10log10 rj 10log10sj) 2σ 2 (4.5) 2πǫr j where M denotes the specific choice of signal model and noise statistics, θ = (T x,t y ) is the unknown emitter s location parameter and r = [r 1 r 2... r N ] are the sensors power 55

68 y [m] x [m] Figure 4.5: CRLB for uniform sensor placement for all emitter locations (N t = 1, N s = 100, L = 1 km, α = 4, σ = 4 db) measurements. The shadowing spread σ can be considered as a nuisance parameter and can be integrated out analytically to obtain the following marginalized likelihood function, L mar (Refer to Appendix B.1). L mar = p(r M,θ) = K [ Ns j=1 log 2 10 r j d α j ρp t ] Ns 2 (4.6) ( ) where K = 1 Ns ( Ns ) πǫ j=1 r Ns 2 j Γ( Ns ) is a constant parameter (scaling factor), 2 which is independent of emitter power and location. Here Γ(.) represents the gamma function. The advantage of using the marginalized likelihood is that it eliminates the need for knowing or estimating the shadowing spread. Obtaining the exact solution of the ML estimate requires taking the derivative of (4.6) with respect to the unknown parameter which cannot be carried out analytically. A numerical solution to this problem can be obtained by searching the value of the parameter that gives the maximum value of the likelihood function over a grid element, which is called grid-search method. In a grid-search method, the solution space is divided into a number of grids and a cost function is calculated for each grid center. The cost function that gives the estimate of the location parameter, ˆθ = (ˆT x, ˆT y ) can be obtained by minimizing the sum of squared 56

69 differences between each sensor s power estimate and average of all sensor estimates for each grid center [82]. Note that the emitter power is not known and the value of the cost function depends on the power estimates of each sensor in the network. The center of the grid that minimizes the cost function is selected as the solution: ˆθ = min T x, T y N s j=1 ( log(r j d α T,j ) 1 N s N s i=1 log(r i d α T,i ) ) 2 (4.7) where d T,j is the distance between jth sensor and the test grid location T. The derivation of the cost function in (4.7) is given in [82]. As can be seen from (4.7), the values of the transmit power and the shadowing spread are not required for cost function calculations. Thus, the MLE method is advantageous for the single emitter case scenarios where the emitter power and db spread parameters are not known. A sample grid is illustrated in Fig. 4.6 to show how the technique is used in estimating the locationof a single emitter using five sensors. In this example, 1 km by 1 km simulation region is divided into smaller squares of length 100 m. The total number of grid elements Q, that is required to search is 100. The sensors are shown with circles whereas the actual emitter location is marked with a diamond shape. The emitter power is assumed to be unknown, however the received power measurements at sensor locations are generated due to 1 W transmit power for our simulations. The values of the cost function calculated in each grid location are also shown in the figure. As seen from the figure, the minimum value of the cost function which is 3 for this example, is obtained around the actual emitter location. The estimated emitter location is shown with a square at the center of the grid for which the cost is minimum. In order to compare the performance of grid search based ML estimator to the CRLB, two different emitter-sensor geometries have been considered. In both cases, we assume independent log-normal shadowing of given db spread and log-distance path-loss model with a fixed path-loss exponent of 4. The emitter power is not known and the grid-size is set to 10 m by 10 m. In the first case, the emitter is located at the center of the area of 1200 m by 1200 m and 6 sensors are evenly positioned on the circle centered at the emitter with a radius of 500 m as shown in Fig. 4.7 for comparison to [78]. The RMS of estimation error is plotted for db spread values of 1 to 12 db for 1000 simulations in Fig As can be seen from the figure, the performance of the full grid search based on the ML approach is close to the Cramer-Rao lower bound for shadowing spread values of 6 db and under. In the second case, a network of 100 sensors distributed placed in a square lattice with uniform spacing between them in a 1 km by 1 km area and a primary emitter located at the center of the area as in Fig The RMS of estimation error is plotted for db spread values of 1 to 12 db for 100 simulations in Fig As seen from Fig. 4.9, the performance of the ML estimator approaches to CRLB when the shadowing spread values are between 3 db and 9 db. However the performance degrades with higher db spread values since higher shadowing effect degrades the estimation 57

70 Figure 4.6: Uniform grid structure for full search (N t = 1, N s =5, L = 1 km, grid size = 100 m by 100 m, P t = 1 W) Figure 4.7: Sensors positioned on the circle centered at the emitter 58

71 Grid search CRLB RMSE [m] Shadow spread (db) Figure 4.8: Performance of full grid search compared to CRLB for six sensors that are evenly positioned on the circle centered at the emitter with a radius of 500 m (N t = 1, N s = 6, L = 1200 m, α = 4, grid size = 10 m by 10 m, P t = 1 W) Grid search CRLB RMSE [m] X: 0 Y: Shadow spread (db) Figure 4.9: CRLB and the RMSE of the grid search for uniform sensor placement (N t = 1, N s = 100, grid size = 10 m by 10 m, L = 1 km, α = 4, P t = 1 W) 59

72 performance. Moreover, the quantization noise due to selecting large grid sizes becomes more pronounced than the shadowing effect below 4 db, therefore deviations from the CRLB is observed at smaller shadowing spread values. When there is no shadowing, the RMS of emitter localization error with the grid search method is calculated as for a grid size of 10 m and it is depicted in the figure as (X:0, Y:7.071). This is due to the position of emitter being in the corner of the grid where it resides as in Fig. 4.2 and its estimated location is found to be in the center of that grid. Therefore the distance between the corner and the center of a grid which is gridsize 2 2, determines the quantization error for that particular emitter-sensor geometry Computational Complexity of Grid Search Method The RMSE of the MLE method based on full grid search closely achieves the Cramer-Rao lower bound at small shadowing variances as it is shown in the previous section. However, full grid search is a complex search method that requires the algorithm to scan all possible grid points in the localization space and also requires the size of the grid elements to be chosen small to reduce the quantization error induced by the grid size and obtain more accurate location estimates. There is a trade-off between the estimation accuracy and the computational complexity; smaller grid sizes provide more accurate estimates at a cost of increased computational complexity. The algorithm becomes even more complex for multiple primary emitter localization problems as it increases the calculation of the likelihood function for all grid points related with the number of primary emitters considered in the network. The number of computations is given by; ( ) Q Number of computations = N t (4.8) where Q is the number of grid locations and N t is the number of emitters. For example for two emitter scenario, the number of calculations is Q(Q 1)/2 where in one emitter case it is Q that is the total number of grid elements. The number of computations versus the number of emitters is plotted in Fig As can be seen from the figure, the complexity increases exponentially with respect to the number of unknown parameters to be estimated. In the case of estimating multiple parameters such as emitter location, emitter power, path-loss exponent, joint parameter estimation is not possible with the grid search method. Instead, each parameter is estimated one at a time. This situation yields in accumulating estimation errors for the last estimated parameters. For example, for a single emitter case scenario, where the problem is to estimate both the transmit power and its location, the error that arises from location estimation will affect the transmit power estimate. Therefore for multiple parameter estimation problems, this method is not preferred. Additionally, any prior knowledge about the location of the transmitter or any statistical characterization of its transmit power is not considered with the ML estimator, which is called as blind estimation. 60

73 Number of computations Number of emitters Figure 4.10: Computational complexity of the MLE method for Q = Single Emitter Localization Using MCMC Method The problems with the grid search method that are discussed in the previous section, arises the need to look for another numerical solution that can also account for the shadowing uncertainty introduced by the wireless channel. In this section, we will look into numerical approaches based on generating samples from the joint posterior of the unknown parameters with MCMC methods, as an alternative for highly computationally complex full grid search method. The reason to propose a MCMC method for maximizing the likelihood function given in (4.6) is that the MCMC methods provide computationally efficient sampling strategies especially when the number of parameters to be estimated is large, which is the case for multiple emitter scenarios. Besides, the MCMC estimator has a Bayesian approach in which a prior knowledge can be used for better estimates. The rough emitter location estimates we have obtained from generated REM in the previous chapter is taken as a prior knowledge and it is proposed to initialize the MCMC algorithm for reducing the number of iterations required for convergence as described in Section 5.2. Moreover, the quantization error we have encountered in the grid search technique is not a concern with any of the MCMC methods. Basically, for maximizing the likelihood function given in (4.6), samples are generated from this likelihood function using an MCMC thechnique. The simulated samples generated by this process are then used for making inferences about the emitter s location. 61

74 Metropolis Hastings (MH) Algorithm MCMC methods are a class of iterative algorithms for sampling from probability distributions based on constructing a Markov chain that has the desired distribution as its equilibrium distribution [60]. The state of the chain after a large number of steps is then used as a sample of the desired distribution. The quality of the sample improves as a function of the number of iterations. Samples of the Markov chain are generated by using Metropolis Hasting (MH) algorithm [17], an MCMC method that is based on Accept- Reject technique. As a motivation for the MH algorithm, we first present the theory that lies behind the Accept -Reject idea. Simulation methods that are for pseudo-random number sampling are based on the production of uniform random variables, originally independent random variables that are distributed according to a distribution p, that is not necessarily explicitly known. The uniform distribution U(0, 1) provides the basic probabilistic representation of randomness and the generation of uniform random variables is the key point in the behavior of simulation methods for other probability distributions since those distributions can be represented as a deterministic transformation of uniform random variables. For example, inverse transform sampling, also known as inverse transform method is one of the basic methods for pseudo-random number sampling, i.e. for generating sample numbers at random (random variables) from any probability distribution given its invertible cumulative distribution function (cdf) [60]. Similarly, when a distribution p is linked in a relatively simple way to another distribution that is easy to simulate, this relationship can often be exploited to construct an algorithm to simulate variables from p. There are many transformation algorithms for generating non uniform random variables using relationships between the distributions. For example Box-Muller algorithm is used for normal variable generation connected to the uniform distribution. Similarly, Poisson distribution can be simulated by generating exponential random variables. However there are many distributions from which it is difficult, or even impossible to directly simulate by an inverse transform. Moreover, in some cases, we are not even able to represent the distribution in a usable form such as transformation. In such settings, it is impossible to exploit direct probabilistic probabilities to derive a simulation method. We thus turn to another class of methods that only requires us to know the functional form of the density p of interest up to a multiplicative constant; no deep analytical study of p is necessary. The key to this method is to use a simpler (simulation wise) density q from which the simulation is actually done. For a given density q, called the instrumental density, there are thus many densities p, called the target densities, which can be simulated this way. The corresponding algorithm, called Accept-Reject, is based on a simple connection with the uniform distribution and can be explained as follows. Consider an envelope distribution as an instrumental distribution around the underlying target distribution p(x). In the simplest case it can be a uniform distribution around it. Then choose (x,y) such that: x U(A,B) y U(0,max(f(x))) If y is under the p(x) curve accept x, otherwise reject it. An example can be found in Fig [13]. 62

75 Figure 4.11: An example for Accept-Reject technique, uniform envelope Figure 4.12: An example for Accept-Reject technique, scaled Gaussian envelope 63